id	sid	tid	token	lemma	pos
ejpam-4854	1	1	european	european	PROPN
ejpam-4854	1	2	journal	journal	PROPN
ejpam-4854	1	3	of	of	ADP
ejpam-4854	1	4	pure	pure	ADJ
ejpam-4854	1	5	and	and	CCONJ
ejpam-4854	1	6	applied	apply	VERB
ejpam-4854	1	7	mathematics	mathematic	NOUN
ejpam-4854	1	8	vol	vol	NOUN
ejpam-4854	1	9	.	.	PUNCT
ejpam-4854	2	1	16	16	NUM
ejpam-4854	2	2	,	,	PUNCT
ejpam-4854	2	3	no	no	INTJ
ejpam-4854	2	4	.	.	NOUN
ejpam-4854	2	5	4	4	NUM
ejpam-4854	2	6	,	,	PUNCT
ejpam-4854	2	7	2023	2023	NUM
ejpam-4854	2	8	,	,	PUNCT
ejpam-4854	2	9	2306	2306	NUM
ejpam-4854	2	10	-	-	SYM
ejpam-4854	2	11	2322	2322	NUM
ejpam-4854	2	12	issn	issn	PROPN
ejpam-4854	2	13	1307	1307	NUM
ejpam-4854	2	14	-	-	SYM
ejpam-4854	2	15	5543	5543	NUM
ejpam-4854	2	16	–	–	PUNCT
ejpam-4854	2	17	ejpam.com	ejpam.com	X
ejpam-4854	2	18	published	publish	VERB
ejpam-4854	2	19	by	by	ADP
ejpam-4854	2	20	new	new	PROPN
ejpam-4854	2	21	york	york	PROPN
ejpam-4854	2	22	business	business	PROPN
ejpam-4854	2	23	global	global	PROPN
ejpam-4854	2	24	the	the	DET
ejpam-4854	2	25	linear	linear	ADJ
ejpam-4854	2	26	algebra	algebra	NOUN
ejpam-4854	2	27	of	of	ADP
ejpam-4854	2	28	the	the	DET
ejpam-4854	2	29	(	(	PUNCT
ejpam-4854	2	30	r	r	NOUN
ejpam-4854	2	31	,	,	PUNCT
ejpam-4854	2	32	β)-stirling	β)-stirle	VERB
ejpam-4854	2	33	matrices	matrix	NOUN
ejpam-4854	2	34	genevieve	genevieve	PROPN
ejpam-4854	2	35	b.	b.	PROPN
ejpam-4854	2	36	engalan1,∗	engalan1,∗	PROPN
ejpam-4854	2	37	,	,	PUNCT
ejpam-4854	2	38	mary	mary	PROPN
ejpam-4854	2	39	joy	joy	PROPN
ejpam-4854	2	40	r.	r.	PROPN
ejpam-4854	2	41	latayada1	latayada1	PROPN
ejpam-4854	2	42	1	1	NUM
ejpam-4854	2	43	department	department	NOUN
ejpam-4854	2	44	of	of	ADP
ejpam-4854	2	45	mathematics	mathematic	NOUN
ejpam-4854	2	46	,	,	PUNCT
ejpam-4854	2	47	caraga	caraga	PROPN
ejpam-4854	2	48	state	state	PROPN
ejpam-4854	2	49	university	university	PROPN
ejpam-4854	2	50	,	,	PUNCT
ejpam-4854	2	51	butuan	butuan	PROPN
ejpam-4854	2	52	city	city	PROPN
ejpam-4854	2	53	,	,	PUNCT
ejpam-4854	2	54	philippines	philippine	NOUN
ejpam-4854	2	55	abstract	abstract	ADJ
ejpam-4854	2	56	.	.	PUNCT
ejpam-4854	3	1	this	this	DET
ejpam-4854	3	2	paper	paper	NOUN
ejpam-4854	3	3	establishes	establish	VERB
ejpam-4854	3	4	the	the	DET
ejpam-4854	3	5	linear	linear	ADJ
ejpam-4854	3	6	algebra	algebra	NOUN
ejpam-4854	3	7	of	of	ADP
ejpam-4854	3	8	the	the	DET
ejpam-4854	3	9	(	(	PUNCT
ejpam-4854	3	10	r	r	NOUN
ejpam-4854	3	11	,	,	PUNCT
ejpam-4854	3	12	β)-stirling	β)-stirle	VERB
ejpam-4854	3	13	matrix	matrix	NOUN
ejpam-4854	3	14	.	.	PUNCT
ejpam-4854	4	1	along	along	ADP
ejpam-4854	4	2	the	the	DET
ejpam-4854	4	3	way	way	NOUN
ejpam-4854	4	4	,	,	PUNCT
ejpam-4854	4	5	this	this	DET
ejpam-4854	4	6	paper	paper	NOUN
ejpam-4854	4	7	derives	derive	VERB
ejpam-4854	4	8	various	various	ADJ
ejpam-4854	4	9	identities	identity	NOUN
ejpam-4854	4	10	,	,	PUNCT
ejpam-4854	4	11	such	such	ADJ
ejpam-4854	4	12	as	as	ADP
ejpam-4854	4	13	its	its	PRON
ejpam-4854	4	14	factorization	factorization	NOUN
ejpam-4854	4	15	and	and	CCONJ
ejpam-4854	4	16	relationship	relationship	NOUN
ejpam-4854	4	17	to	to	ADP
ejpam-4854	4	18	the	the	DET
ejpam-4854	4	19	pascal	pascal	ADJ
ejpam-4854	4	20	matrix	matrix	NOUN
ejpam-4854	4	21	and	and	CCONJ
ejpam-4854	4	22	the	the	DET
ejpam-4854	4	23	stirling	stirling	NOUN
ejpam-4854	4	24	matrix	matrix	NOUN
ejpam-4854	4	25	of	of	ADP
ejpam-4854	4	26	the	the	DET
ejpam-4854	4	27	second	second	ADJ
ejpam-4854	4	28	kind	kind	NOUN
ejpam-4854	4	29	.	.	PUNCT
ejpam-4854	5	1	additionally	additionally	ADV
ejpam-4854	5	2	,	,	PUNCT
ejpam-4854	5	3	this	this	DET
ejpam-4854	5	4	paper	paper	NOUN
ejpam-4854	5	5	develops	develop	VERB
ejpam-4854	5	6	a	a	DET
ejpam-4854	5	7	natural	natural	ADJ
ejpam-4854	5	8	extension	extension	NOUN
ejpam-4854	5	9	of	of	ADP
ejpam-4854	5	10	the	the	DET
ejpam-4854	5	11	vandermonde	vandermonde	ADJ
ejpam-4854	5	12	matrix	matrix	NOUN
ejpam-4854	5	13	,	,	PUNCT
ejpam-4854	5	14	which	which	PRON
ejpam-4854	5	15	can	can	AUX
ejpam-4854	5	16	be	be	AUX
ejpam-4854	5	17	used	use	VERB
ejpam-4854	5	18	to	to	PART
ejpam-4854	5	19	study	study	VERB
ejpam-4854	5	20	and	and	CCONJ
ejpam-4854	5	21	evaluate	evaluate	VERB
ejpam-4854	5	22	successive	successive	ADJ
ejpam-4854	5	23	power	power	NOUN
ejpam-4854	5	24	sums	sum	NOUN
ejpam-4854	5	25	of	of	ADP
ejpam-4854	5	26	arithmetic	arithmetic	ADJ
ejpam-4854	5	27	progressions	progression	NOUN
ejpam-4854	5	28	.	.	PUNCT
ejpam-4854	6	1	2020	2020	NUM
ejpam-4854	6	2	mathematics	mathematic	NOUN
ejpam-4854	6	3	subject	subject	NOUN
ejpam-4854	6	4	classifications	classification	NOUN
ejpam-4854	6	5	:	:	PUNCT
ejpam-4854	6	6	05	05	NUM
ejpam-4854	6	7	,	,	PUNCT
ejpam-4854	6	8	11	11	NUM
ejpam-4854	6	9	key	key	ADJ
ejpam-4854	6	10	words	word	NOUN
ejpam-4854	6	11	and	and	CCONJ
ejpam-4854	6	12	phrases	phrase	NOUN
ejpam-4854	6	13	:	:	PUNCT
ejpam-4854	6	14	(	(	PUNCT
ejpam-4854	6	15	r	r	NOUN
ejpam-4854	6	16	,	,	PUNCT
ejpam-4854	6	17	β)-stirling	β)-stirle	VERB
ejpam-4854	6	18	numbers	number	NOUN
ejpam-4854	6	19	,	,	PUNCT
ejpam-4854	6	20	(	(	PUNCT
ejpam-4854	6	21	r	r	NOUN
ejpam-4854	6	22	,	,	PUNCT
ejpam-4854	6	23	β)-stirling	β)-stirle	VERB
ejpam-4854	6	24	matrix	matrix	NOUN
ejpam-4854	6	25	,	,	PUNCT
ejpam-4854	6	26	pascal	pascal	ADJ
ejpam-4854	6	27	matrix	matrix	NOUN
ejpam-4854	6	28	,	,	PUNCT
ejpam-4854	6	29	vandermonde	vandermonde	ADJ
ejpam-4854	6	30	matrix	matrix	NOUN
ejpam-4854	6	31	1	1	NUM
ejpam-4854	6	32	.	.	PUNCT
ejpam-4854	6	33	introduction	introduction	NOUN
ejpam-4854	6	34	the	the	DET
ejpam-4854	6	35	(	(	PUNCT
ejpam-4854	6	36	r	r	NOUN
ejpam-4854	6	37	,	,	PUNCT
ejpam-4854	6	38	β)-stirling	β)-stirling	VERB
ejpam-4854	6	39	numbers	number	NOUN
ejpam-4854	6	40	are	be	AUX
ejpam-4854	6	41	a	a	DET
ejpam-4854	6	42	generalization	generalization	NOUN
ejpam-4854	6	43	of	of	ADP
ejpam-4854	6	44	the	the	DET
ejpam-4854	6	45	classical	classical	ADJ
ejpam-4854	6	46	stirling	stirling	NOUN
ejpam-4854	6	47	numbers	number	NOUN
ejpam-4854	6	48	of	of	ADP
ejpam-4854	6	49	the	the	DET
ejpam-4854	6	50	second	second	ADJ
ejpam-4854	6	51	kind	kind	NOUN
ejpam-4854	6	52	and	and	CCONJ
ejpam-4854	6	53	r	r	NOUN
ejpam-4854	6	54	-	-	PUNCT
ejpam-4854	6	55	stirling	stirling	NOUN
ejpam-4854	6	56	numbers	number	NOUN
ejpam-4854	6	57	,	,	PUNCT
ejpam-4854	6	58	and	and	CCONJ
ejpam-4854	6	59	is	be	AUX
ejpam-4854	6	60	denoted	denote	VERB
ejpam-4854	6	61	by	by	ADP
ejpam-4854	6	62	,	,	PUNCT
ejpam-4854	6	63	〈	〈	PROPN
ejpam-4854	6	64	n	n	ADV
ejpam-4854	6	65	k	k	X
ejpam-4854	6	66	〉	〉	X
ejpam-4854	6	67	β	β	NOUN
ejpam-4854	6	68	,	,	PUNCT
ejpam-4854	6	69	r	r	NOUN
ejpam-4854	6	70	.	.	PUNCT
ejpam-4854	7	1	they	they	PRON
ejpam-4854	7	2	were	be	AUX
ejpam-4854	7	3	introduced	introduce	VERB
ejpam-4854	7	4	by	by	ADP
ejpam-4854	7	5	corcino	corcino	NOUN
ejpam-4854	7	6	in	in	ADP
ejpam-4854	7	7	1999	1999	NUM
ejpam-4854	7	8	[	[	X
ejpam-4854	7	9	5	5	NUM
ejpam-4854	7	10	]	]	PUNCT
ejpam-4854	7	11	by	by	ADP
ejpam-4854	7	12	means	mean	NOUN
ejpam-4854	7	13	of	of	ADP
ejpam-4854	7	14	the	the	DET
ejpam-4854	7	15	following	following	ADJ
ejpam-4854	7	16	linear	linear	PROPN
ejpam-4854	7	17	transformation	transformation	NOUN
ejpam-4854	7	18	:	:	PUNCT
ejpam-4854	7	19	tn	tn	PROPN
ejpam-4854	7	20	=	=	SYM
ejpam-4854	7	21	n∑	n∑	PROPN
ejpam-4854	7	22	k=0	k=0	PROPN
ejpam-4854	7	23	〈	〈	PROPN
ejpam-4854	7	24	n	n	CCONJ
ejpam-4854	7	25	k	k	X
ejpam-4854	7	26	〉	〉	X
ejpam-4854	7	27	β	β	X
ejpam-4854	7	28	,	,	PUNCT
ejpam-4854	7	29	r	r	NOUN
ejpam-4854	7	30	(	(	PUNCT
ejpam-4854	7	31	t−	t−	PRON
ejpam-4854	7	32	r)β	r)β	NOUN
ejpam-4854	7	33	,	,	PUNCT
ejpam-4854	7	34	k	k	PROPN
ejpam-4854	7	35	(	(	PUNCT
ejpam-4854	7	36	1	1	NUM
ejpam-4854	7	37	)	)	PUNCT
ejpam-4854	7	38	where	where	SCONJ
ejpam-4854	7	39	(	(	PUNCT
ejpam-4854	7	40	t−	t−	PRON
ejpam-4854	7	41	r)β	r)β	NOUN
ejpam-4854	7	42	,	,	PUNCT
ejpam-4854	7	43	k	k	PROPN
ejpam-4854	7	44	=	=	SYM
ejpam-4854	7	45	k−1∏	k−1∏	PROPN
ejpam-4854	7	46	i=0	i=0	PROPN
ejpam-4854	7	47	(	(	PUNCT
ejpam-4854	7	48	t−	t−	PROPN
ejpam-4854	7	49	r	r	NOUN
ejpam-4854	7	50	−	−	NOUN
ejpam-4854	7	51	iβ	iβ	NOUN
ejpam-4854	7	52	)	)	PUNCT
ejpam-4854	7	53	.	.	PUNCT
ejpam-4854	8	1	(	(	PUNCT
ejpam-4854	8	2	2	2	X
ejpam-4854	8	3	)	)	PUNCT
ejpam-4854	8	4	(	(	PUNCT
ejpam-4854	8	5	t)β	t)β	NOUN
ejpam-4854	8	6	,	,	PUNCT
ejpam-4854	8	7	k	k	PROPN
ejpam-4854	8	8	is	be	AUX
ejpam-4854	8	9	called	call	VERB
ejpam-4854	8	10	the	the	DET
ejpam-4854	8	11	generalized	generalized	ADJ
ejpam-4854	8	12	factorial	factorial	NOUN
ejpam-4854	8	13	of	of	ADP
ejpam-4854	8	14	t	t	PROPN
ejpam-4854	8	15	with	with	ADP
ejpam-4854	8	16	increment	increment	NOUN
ejpam-4854	8	17	β	β	NOUN
ejpam-4854	8	18	,	,	PUNCT
ejpam-4854	8	19	and	and	CCONJ
ejpam-4854	8	20	as	as	ADP
ejpam-4854	8	21	a	a	DET
ejpam-4854	8	22	convention	convention	NOUN
ejpam-4854	8	23	(	(	PUNCT
ejpam-4854	8	24	t)β	t)β	NOUN
ejpam-4854	8	25	,	,	PUNCT
ejpam-4854	8	26	k	k	PROPN
ejpam-4854	8	27	=	=	PUNCT
ejpam-4854	8	28	0	0	PUNCT
ejpam-4854	9	1	if	if	SCONJ
ejpam-4854	9	2	k	k	PROPN
ejpam-4854	9	3	≤	≤	ADV
ejpam-4854	9	4	0	0	NUM
ejpam-4854	9	5	.	.	PUNCT
ejpam-4854	10	1	this	this	DET
ejpam-4854	10	2	numbers	number	NOUN
ejpam-4854	10	3	have	have	VERB
ejpam-4854	10	4	applications	application	NOUN
ejpam-4854	10	5	in	in	ADP
ejpam-4854	10	6	combinatorial	combinatorial	ADJ
ejpam-4854	10	7	and	and	CCONJ
ejpam-4854	10	8	statistical	statistical	ADJ
ejpam-4854	10	9	problems	problem	NOUN
ejpam-4854	10	10	.	.	PUNCT
ejpam-4854	11	1	corcino	corcino	NOUN
ejpam-4854	11	2	and	and	CCONJ
ejpam-4854	11	3	aldema	aldema	PROPN
ejpam-4854	11	4	(	(	PUNCT
ejpam-4854	11	5	2002	2002	NUM
ejpam-4854	11	6	)	)	PUNCT
ejpam-4854	12	1	[	[	X
ejpam-4854	12	2	6	6	NUM
ejpam-4854	12	3	]	]	PUNCT
ejpam-4854	12	4	further	far	ADV
ejpam-4854	12	5	studied	study	VERB
ejpam-4854	12	6	the	the	DET
ejpam-4854	12	7	(	(	PUNCT
ejpam-4854	12	8	r	r	NOUN
ejpam-4854	12	9	,	,	PUNCT
ejpam-4854	12	10	β)-stirling	β)-stirle	VERB
ejpam-4854	12	11	numbers	number	NOUN
ejpam-4854	12	12	and	and	CCONJ
ejpam-4854	12	13	derived	derive	VERB
ejpam-4854	12	14	some	some	DET
ejpam-4854	12	15	combinatorial	combinatorial	ADJ
ejpam-4854	12	16	identities	identity	NOUN
ejpam-4854	12	17	related	relate	VERB
ejpam-4854	12	18	to	to	ADP
ejpam-4854	12	19	them	they	PRON
ejpam-4854	12	20	.	.	PUNCT
ejpam-4854	13	1	corcino	corcino	NOUN
ejpam-4854	13	2	and	and	CCONJ
ejpam-4854	13	3	montero	montero	PROPN
ejpam-4854	13	4	(	(	PUNCT
ejpam-4854	13	5	2009	2009	NUM
ejpam-4854	13	6	)	)	PUNCT
ejpam-4854	14	1	[	[	X
ejpam-4854	14	2	7	7	X
ejpam-4854	14	3	]	]	PUNCT
ejpam-4854	14	4	also	also	ADV
ejpam-4854	14	5	investigated	investigate	VERB
ejpam-4854	14	6	the	the	DET
ejpam-4854	14	7	(	(	PUNCT
ejpam-4854	14	8	r	r	NOUN
ejpam-4854	14	9	,	,	PUNCT
ejpam-4854	14	10	β)-stirling	β)-stirle	VERB
ejpam-4854	14	11	numbers	number	NOUN
ejpam-4854	14	12	in	in	ADP
ejpam-4854	14	13	the	the	DET
ejpam-4854	14	14	context	context	NOUN
ejpam-4854	14	15	of	of	ADP
ejpam-4854	14	16	0	0	NUM
ejpam-4854	14	17	-	-	SYM
ejpam-4854	14	18	1	1	NUM
ejpam-4854	14	19	tableaux	tableau	NOUN
ejpam-4854	14	20	,	,	PUNCT
ejpam-4854	14	21	which	which	PRON
ejpam-4854	14	22	are	be	AUX
ejpam-4854	14	23	a	a	DET
ejpam-4854	14	24	tool	tool	NOUN
ejpam-4854	14	25	used	use	VERB
ejpam-4854	14	26	in	in	ADP
ejpam-4854	14	27	algebraic	algebraic	ADJ
ejpam-4854	14	28	combinatorics	combinatoric	NOUN
ejpam-4854	14	29	.	.	PUNCT
ejpam-4854	15	1	∗corresponding	∗corresponde	VERB
ejpam-4854	15	2	author	author	NOUN
ejpam-4854	15	3	.	.	PUNCT
ejpam-4854	16	1	doi	doi	NOUN
ejpam-4854	16	2	:	:	PUNCT
ejpam-4854	16	3	https://doi.org/10.29020/nybg.ejpam.v16i4.4854	https://doi.org/10.29020/nybg.ejpam.v16i4.4854	PROPN
ejpam-4854	16	4	email	email	NOUN
ejpam-4854	16	5	addresses	address	NOUN
ejpam-4854	16	6	:	:	PUNCT
ejpam-4854	16	7	gequisto@carsu.edu.ph	gequisto@carsu.edu.ph	PROPN
ejpam-4854	16	8	(	(	PUNCT
ejpam-4854	16	9	g.	g.	PROPN
ejpam-4854	16	10	engalan	engalan	PROPN
ejpam-4854	16	11	)	)	PUNCT
ejpam-4854	16	12	,	,	PUNCT
ejpam-4854	16	13	mrlatayada@carsu.edu.ph	mrlatayada@carsu.edu.ph	PROPN
ejpam-4854	16	14	(	(	PUNCT
ejpam-4854	16	15	m.j	m.j	PROPN
ejpam-4854	16	16	.	.	PROPN
ejpam-4854	16	17	layatada	layatada	PROPN
ejpam-4854	16	18	)	)	PUNCT
ejpam-4854	16	19	https://www.ejpam.com	https://www.ejpam.com	X
ejpam-4854	16	20	2306	2306	NUM
ejpam-4854	17	1	©	©	PROPN
ejpam-4854	17	2	2023	2023	NUM
ejpam-4854	17	3	ejpam	ejpam	NOUN
ejpam-4854	17	4	all	all	DET
ejpam-4854	17	5	rights	right	NOUN
ejpam-4854	17	6	reserved	reserve	VERB
ejpam-4854	17	7	.	.	PUNCT
ejpam-4854	18	1	g.	g.	PROPN
ejpam-4854	18	2	engalan	engalan	PROPN
ejpam-4854	18	3	,	,	PUNCT
ejpam-4854	18	4	m.r	m.r	PROPN
ejpam-4854	18	5	.	.	PROPN
ejpam-4854	18	6	latayada	latayada	PROPN
ejpam-4854	18	7	/	/	SYM
ejpam-4854	18	8	eur	eur	PROPN
ejpam-4854	18	9	.	.	PUNCT
ejpam-4854	19	1	j.	j.	PROPN
ejpam-4854	19	2	pure	pure	PROPN
ejpam-4854	19	3	appl	appl	PROPN
ejpam-4854	19	4	.	.	PROPN
ejpam-4854	19	5	math	math	PROPN
ejpam-4854	19	6	,	,	PUNCT
ejpam-4854	19	7	16	16	NUM
ejpam-4854	19	8	(	(	PUNCT
ejpam-4854	19	9	4	4	NUM
ejpam-4854	19	10	)	)	PUNCT
ejpam-4854	19	11	(	(	PUNCT
ejpam-4854	19	12	2023	2023	NUM
ejpam-4854	19	13	)	)	PUNCT
ejpam-4854	19	14	,	,	PUNCT
ejpam-4854	19	15	2306	2306	NUM
ejpam-4854	19	16	-	-	SYM
ejpam-4854	19	17	2322	2322	NUM
ejpam-4854	19	18	2307	2307	NUM
ejpam-4854	19	19	in	in	ADP
ejpam-4854	19	20	this	this	DET
ejpam-4854	19	21	paper	paper	NOUN
ejpam-4854	19	22	,	,	PUNCT
ejpam-4854	19	23	we	we	PRON
ejpam-4854	19	24	introduce	introduce	VERB
ejpam-4854	19	25	and	and	CCONJ
ejpam-4854	19	26	study	study	VERB
ejpam-4854	19	27	the	the	DET
ejpam-4854	19	28	(	(	PUNCT
ejpam-4854	19	29	r	r	NOUN
ejpam-4854	19	30	,	,	PUNCT
ejpam-4854	19	31	β)-stirling	β)-stirle	VERB
ejpam-4854	19	32	matrix	matrix	NOUN
ejpam-4854	19	33	and	and	CCONJ
ejpam-4854	19	34	we	we	PRON
ejpam-4854	19	35	derive	derive	VERB
ejpam-4854	19	36	several	several	ADJ
ejpam-4854	19	37	interesting	interesting	ADJ
ejpam-4854	19	38	identities	identity	NOUN
ejpam-4854	19	39	about	about	ADP
ejpam-4854	19	40	(	(	PUNCT
ejpam-4854	19	41	r	r	NOUN
ejpam-4854	19	42	,	,	PUNCT
ejpam-4854	19	43	β)-stirling	β)-stirle	VERB
ejpam-4854	19	44	sequence	sequence	NOUN
ejpam-4854	19	45	.	.	PUNCT
ejpam-4854	20	1	two	two	NUM
ejpam-4854	20	2	applications	application	NOUN
ejpam-4854	20	3	are	be	AUX
ejpam-4854	20	4	given	give	VERB
ejpam-4854	20	5	:	:	PUNCT
ejpam-4854	20	6	we	we	PRON
ejpam-4854	20	7	can	can	AUX
ejpam-4854	20	8	generalize	generalize	VERB
ejpam-4854	20	9	the	the	DET
ejpam-4854	20	10	vandermonde	vandermonde	ADJ
ejpam-4854	20	11	matrices	matrix	NOUN
ejpam-4854	20	12	and	and	CCONJ
ejpam-4854	20	13	evaluate	evaluate	VERB
ejpam-4854	20	14	successive	successive	ADJ
ejpam-4854	20	15	power	power	NOUN
ejpam-4854	20	16	sums	sum	NOUN
ejpam-4854	20	17	of	of	ADP
ejpam-4854	20	18	arithmetic	arithmetic	ADJ
ejpam-4854	20	19	progressions	progression	NOUN
ejpam-4854	20	20	.	.	PUNCT
ejpam-4854	21	1	2	2	X
ejpam-4854	21	2	.	.	X
ejpam-4854	21	3	results	result	VERB
ejpam-4854	21	4	2.1	2.1	NUM
ejpam-4854	21	5	.	.	PUNCT
ejpam-4854	22	1	the	the	DET
ejpam-4854	22	2	(	(	PUNCT
ejpam-4854	22	3	r	r	NOUN
ejpam-4854	22	4	,	,	PUNCT
ejpam-4854	22	5	β)-stirling	β)-stirle	VERB
ejpam-4854	22	6	matrix	matrix	NOUN
ejpam-4854	22	7	the	the	DET
ejpam-4854	22	8	key	key	ADJ
ejpam-4854	22	9	notions	notion	NOUN
ejpam-4854	22	10	of	of	ADP
ejpam-4854	22	11	the	the	DET
ejpam-4854	22	12	study	study	NOUN
ejpam-4854	22	13	are	be	AUX
ejpam-4854	22	14	now	now	ADV
ejpam-4854	22	15	defined	define	VERB
ejpam-4854	22	16	.	.	PUNCT
ejpam-4854	23	1	definition	definition	NOUN
ejpam-4854	23	2	1	1	NUM
ejpam-4854	23	3	.	.	PUNCT
ejpam-4854	24	1	the	the	DET
ejpam-4854	24	2	(	(	PUNCT
ejpam-4854	24	3	r	r	NOUN
ejpam-4854	24	4	,	,	PUNCT
ejpam-4854	24	5	β)-stirling	β)-stirle	VERB
ejpam-4854	24	6	matrix	matrix	NOUN
ejpam-4854	24	7	is	be	AUX
ejpam-4854	24	8	the	the	DET
ejpam-4854	24	9	n×	n×	PROPN
ejpam-4854	24	10	n	n	NOUN
ejpam-4854	24	11	matrix	matrix	NOUN
ejpam-4854	24	12	defined	define	VERB
ejpam-4854	24	13	by	by	ADP
ejpam-4854	24	14	s(β	s(β	PROPN
ejpam-4854	24	15	,	,	PUNCT
ejpam-4854	24	16	r)(n	r)(n	PROPN
ejpam-4854	24	17	)	)	PUNCT
ejpam-4854	24	18	=	=	PUNCT
ejpam-4854	25	1	[	[	X
ejpam-4854	25	2	〈	〈	PROPN
ejpam-4854	25	3	n	n	ADV
ejpam-4854	25	4	k	k	X
ejpam-4854	25	5	〉	〉	X
ejpam-4854	25	6	β	β	NOUN
ejpam-4854	25	7	,	,	PUNCT
ejpam-4854	25	8	r	r	NOUN
ejpam-4854	25	9	]	]	PUNCT
ejpam-4854	25	10	0≤i	0≤i	PROPN
ejpam-4854	25	11	,	,	PUNCT
ejpam-4854	25	12	j≤n−1	j≤n−1	PROPN
ejpam-4854	25	13	example	example	NOUN
ejpam-4854	26	1	1	1	NUM
ejpam-4854	26	2	.	.	PUNCT
ejpam-4854	26	3	when	when	SCONJ
ejpam-4854	26	4	n	n	X
ejpam-4854	26	5	=	=	SYM
ejpam-4854	26	6	4	4	NUM
ejpam-4854	26	7	,	,	PUNCT
ejpam-4854	26	8	we	we	PRON
ejpam-4854	26	9	have	have	VERB
ejpam-4854	26	10	s(β	s(β	PROPN
ejpam-4854	26	11	,	,	PUNCT
ejpam-4854	26	12	r)(4	r)(4	NUM
ejpam-4854	26	13	)	)	PUNCT
ejpam-4854	27	1	=	=	SYM
ejpam-4854	27	2			NOUN
ejpam-4854	28	1	1	1	NUM
ejpam-4854	28	2	0	0	NUM
ejpam-4854	28	3	0	0	NUM
ejpam-4854	28	4	0	0	NUM
ejpam-4854	29	1	r	r	NOUN
ejpam-4854	29	2	1	1	NUM
ejpam-4854	29	3	0	0	NUM
ejpam-4854	29	4	0	0	NUM
ejpam-4854	29	5	r2	r2	NOUN
ejpam-4854	29	6	β	β	X
ejpam-4854	29	7	+	+	CCONJ
ejpam-4854	29	8	2r	2r	NUM
ejpam-4854	29	9	1	1	NUM
ejpam-4854	29	10	0	0	NUM
ejpam-4854	29	11	r3	r3	PROPN
ejpam-4854	29	12	β2	β2	NOUN
ejpam-4854	29	13	+	+	CCONJ
ejpam-4854	29	14	3βr	3βr	ADJ
ejpam-4854	29	15	+	+	CCONJ
ejpam-4854	29	16	3r2	3r2	NUM
ejpam-4854	29	17	3β	3β	NUM
ejpam-4854	29	18	+	+	CCONJ
ejpam-4854	29	19	3r	3r	NUM
ejpam-4854	29	20	1	1	NUM
ejpam-4854	29	21			NOUN
ejpam-4854	29	22	.	.	PUNCT
ejpam-4854	30	1	(	(	PUNCT
ejpam-4854	30	2	3	3	X
ejpam-4854	30	3	)	)	PUNCT
ejpam-4854	30	4	for	for	ADP
ejpam-4854	30	5	the	the	DET
ejpam-4854	30	6	following	follow	VERB
ejpam-4854	30	7	propositions	proposition	NOUN
ejpam-4854	30	8	,	,	PUNCT
ejpam-4854	30	9	we	we	PRON
ejpam-4854	30	10	need	need	VERB
ejpam-4854	30	11	the	the	DET
ejpam-4854	30	12	generalized	generalized	ADJ
ejpam-4854	30	13	n	n	PRON
ejpam-4854	30	14	×	×	NOUN
ejpam-4854	30	15	n	n	CCONJ
ejpam-4854	30	16	pascal	pascal	ADJ
ejpam-4854	30	17	matrix	matrix	NOUN
ejpam-4854	30	18	and	and	CCONJ
ejpam-4854	30	19	the	the	DET
ejpam-4854	30	20	generalized	generalized	ADJ
ejpam-4854	30	21	n	n	PRON
ejpam-4854	30	22	×	×	NOUN
ejpam-4854	30	23	n	n	CCONJ
ejpam-4854	30	24	stirling	stirling	NOUN
ejpam-4854	30	25	matrix	matrix	NOUN
ejpam-4854	30	26	of	of	ADP
ejpam-4854	30	27	the	the	DET
ejpam-4854	30	28	second	second	ADJ
ejpam-4854	30	29	kind	kind	NOUN
ejpam-4854	30	30	,	,	PUNCT
ejpam-4854	30	31	defined	define	VERB
ejpam-4854	30	32	by	by	ADP
ejpam-4854	30	33	[	[	X
ejpam-4854	30	34	2	2	NUM
ejpam-4854	30	35	]	]	PUNCT
ejpam-4854	30	36	and	and	CCONJ
ejpam-4854	30	37	[	[	X
ejpam-4854	30	38	3	3	NUM
ejpam-4854	30	39	]	]	PUNCT
ejpam-4854	30	40	,	,	PUNCT
ejpam-4854	30	41	respectively	respectively	ADV
ejpam-4854	30	42	as	as	ADP
ejpam-4854	30	43	:	:	PUNCT
ejpam-4854	30	44	pn[x	pn[x	NOUN
ejpam-4854	30	45	]	]	X
ejpam-4854	31	1	=	=	PUNCT
ejpam-4854	31	2	[	[	PUNCT
ejpam-4854	31	3	xi−j	xi−j	PROPN
ejpam-4854	31	4	(	(	PUNCT
ejpam-4854	31	5	i	i	PRON
ejpam-4854	31	6	j	j	PROPN
ejpam-4854	31	7	)	)	PUNCT
ejpam-4854	31	8	]	]	PUNCT
ejpam-4854	32	1	0≤i	0≤i	PROPN
ejpam-4854	32	2	,	,	PUNCT
ejpam-4854	32	3	j≤n−1	j≤n−1	PROPN
ejpam-4854	32	4	where	where	SCONJ
ejpam-4854	32	5	pn	pn	PROPN
ejpam-4854	32	6	=	=	SYM
ejpam-4854	32	7	pn[1	pn[1	PROPN
ejpam-4854	32	8	]	]	X
ejpam-4854	32	9	,	,	PUNCT
ejpam-4854	32	10	and	and	CCONJ
ejpam-4854	32	11	s2(n)[x	s2(n)[x	VERB
ejpam-4854	32	12	]	]	X
ejpam-4854	33	1	=	=	X
ejpam-4854	34	1	[	[	PUNCT
ejpam-4854	34	2	xi−js(i	xi−js(i	PROPN
ejpam-4854	34	3	,	,	PUNCT
ejpam-4854	34	4	j	j	PROPN
ejpam-4854	34	5	)	)	PUNCT
ejpam-4854	34	6	]	]	PUNCT
ejpam-4854	35	1	0≤i	0≤i	PROPN
ejpam-4854	35	2	,	,	PUNCT
ejpam-4854	35	3	j≤n−1	j≤n−1	PROPN
ejpam-4854	35	4	where	where	SCONJ
ejpam-4854	35	5	s(i	s(i	PROPN
ejpam-4854	35	6	,	,	PUNCT
ejpam-4854	35	7	j	j	PROPN
ejpam-4854	35	8	)	)	PUNCT
ejpam-4854	35	9	is	be	AUX
ejpam-4854	35	10	the	the	DET
ejpam-4854	35	11	stirling	stirling	NOUN
ejpam-4854	35	12	numbers	number	NOUN
ejpam-4854	35	13	of	of	ADP
ejpam-4854	35	14	the	the	DET
ejpam-4854	35	15	second	second	ADJ
ejpam-4854	35	16	kind	kind	NOUN
ejpam-4854	35	17	.	.	PUNCT
ejpam-4854	36	1	the	the	DET
ejpam-4854	36	2	main	main	ADJ
ejpam-4854	36	3	technique	technique	NOUN
ejpam-4854	36	4	used	use	VERB
ejpam-4854	36	5	to	to	PART
ejpam-4854	36	6	prove	prove	VERB
ejpam-4854	36	7	the	the	DET
ejpam-4854	36	8	next	next	ADJ
ejpam-4854	36	9	propositions	proposition	NOUN
ejpam-4854	36	10	is	be	AUX
ejpam-4854	36	11	the	the	DET
ejpam-4854	36	12	concept	concept	NOUN
ejpam-4854	36	13	of	of	ADP
ejpam-4854	36	14	the	the	DET
ejpam-4854	36	15	riordan	riordan	PROPN
ejpam-4854	36	16	group	group	NOUN
ejpam-4854	36	17	introduced	introduce	VERB
ejpam-4854	36	18	by	by	ADP
ejpam-4854	36	19	shapiro	shapiro	PROPN
ejpam-4854	36	20	et	et	PROPN
ejpam-4854	36	21	al	al	PROPN
ejpam-4854	36	22	.	.	PUNCT
ejpam-4854	37	1	[	[	X
ejpam-4854	37	2	10	10	NUM
ejpam-4854	37	3	]	]	PUNCT
ejpam-4854	37	4	.	.	PUNCT
ejpam-4854	38	1	this	this	PRON
ejpam-4854	38	2	,	,	PUNCT
ejpam-4854	38	3	briefly	briefly	ADV
ejpam-4854	38	4	,	,	PUNCT
ejpam-4854	38	5	is	be	AUX
ejpam-4854	38	6	a	a	DET
ejpam-4854	38	7	group	group	NOUN
ejpam-4854	38	8	of	of	ADP
ejpam-4854	38	9	infinite	infinite	ADJ
ejpam-4854	38	10	lower	low	ADJ
ejpam-4854	38	11	triangular	triangular	NOUN
ejpam-4854	38	12	arrays	array	NOUN
ejpam-4854	38	13	called	call	VERB
ejpam-4854	38	14	riordan	riordan	PROPN
ejpam-4854	38	15	matrices	matrix	NOUN
ejpam-4854	38	16	.	.	PUNCT
ejpam-4854	39	1	a	a	DET
ejpam-4854	39	2	pair	pair	NOUN
ejpam-4854	39	3	of	of	ADP
ejpam-4854	39	4	formal	formal	ADJ
ejpam-4854	39	5	power	power	NOUN
ejpam-4854	39	6	series	series	PROPN
ejpam-4854	39	7	g	g	PROPN
ejpam-4854	39	8	and	and	CCONJ
ejpam-4854	39	9	f	f	PROPN
ejpam-4854	39	10	in	in	ADP
ejpam-4854	39	11	the	the	DET
ejpam-4854	39	12	ring	ring	NOUN
ejpam-4854	39	13	c[[z	c[[z	PROPN
ejpam-4854	39	14	]	]	X
ejpam-4854	39	15	]	]	PUNCT
ejpam-4854	39	16	define	define	VERB
ejpam-4854	39	17	a	a	DET
ejpam-4854	39	18	riordan	riordan	PROPN
ejpam-4854	39	19	matrix	matrix	NOUN
ejpam-4854	39	20	as	as	ADP
ejpam-4854	39	21	m	m	PROPN
ejpam-4854	39	22	=	=	PROPN
ejpam-4854	40	1	[	[	X
ejpam-4854	40	2	mn	mn	PROPN
ejpam-4854	40	3	,	,	PUNCT
ejpam-4854	40	4	k]n	k]n	PROPN
ejpam-4854	40	5	,	,	PUNCT
ejpam-4854	40	6	k≥0	k≥0	PROPN
ejpam-4854	40	7	,	,	PUNCT
ejpam-4854	40	8	where	where	SCONJ
ejpam-4854	40	9	g(0	g(0	NOUN
ejpam-4854	40	10	)	)	PUNCT
ejpam-4854	40	11	̸=	̸=	PROPN
ejpam-4854	40	12	0	0	NUM
ejpam-4854	40	13	,	,	PUNCT
ejpam-4854	40	14	f(0	f(0	NOUN
ejpam-4854	40	15	)	)	PUNCT
ejpam-4854	40	16	=	=	SYM
ejpam-4854	40	17	0	0	NUM
ejpam-4854	40	18	,	,	PUNCT
ejpam-4854	40	19	f	f	PROPN
ejpam-4854	40	20	′(0	′(0	PROPN
ejpam-4854	40	21	)	)	PUNCT
ejpam-4854	40	22	̸=	̸=	PROPN
ejpam-4854	40	23	0	0	NUM
ejpam-4854	40	24	,	,	PUNCT
ejpam-4854	40	25	and	and	CCONJ
ejpam-4854	40	26	[	[	X
ejpam-4854	40	27	zn	zn	X
ejpam-4854	40	28	]	]	X
ejpam-4854	40	29	is	be	AUX
ejpam-4854	40	30	the	the	DET
ejpam-4854	40	31	coefficient	coefficient	NOUN
ejpam-4854	40	32	extraction	extraction	NOUN
ejpam-4854	40	33	operator	operator	NOUN
ejpam-4854	40	34	.	.	PUNCT
ejpam-4854	41	1	the	the	DET
ejpam-4854	41	2	matrixm	matrixm	NOUN
ejpam-4854	41	3	is	be	AUX
ejpam-4854	41	4	denoted	denote	VERB
ejpam-4854	41	5	by	by	ADP
ejpam-4854	41	6	(	(	PUNCT
ejpam-4854	41	7	g	g	PROPN
ejpam-4854	41	8	,	,	PUNCT
ejpam-4854	41	9	f	f	NOUN
ejpam-4854	41	10	)	)	PUNCT
ejpam-4854	41	11	.	.	PUNCT
ejpam-4854	42	1	moreover	moreover	ADV
ejpam-4854	42	2	,	,	PUNCT
ejpam-4854	42	3	ifmn	ifmn	ADJ
ejpam-4854	42	4	,	,	PUNCT
ejpam-4854	42	5	k	k	X
ejpam-4854	43	1	=	=	PUNCT
ejpam-4854	44	1	[	[	PUNCT
ejpam-4854	44	2	zn	zn	NOUN
ejpam-4854	44	3	n	n	CCONJ
ejpam-4854	44	4	!	!	PUNCT
ejpam-4854	45	1	]	]	X
ejpam-4854	45	2	g	g	X
ejpam-4854	45	3	fk	fk	INTJ
ejpam-4854	45	4	k	k	PROPN
ejpam-4854	45	5	!	!	PUNCT
ejpam-4854	45	6	thenm	thenm	PROPN
ejpam-4854	45	7	is	be	AUX
ejpam-4854	45	8	called	call	VERB
ejpam-4854	45	9	the	the	DET
ejpam-4854	45	10	exponential	exponential	ADJ
ejpam-4854	45	11	riordan	riordan	PROPN
ejpam-4854	45	12	matrix	matrix	NOUN
ejpam-4854	45	13	or	or	CCONJ
ejpam-4854	45	14	e	e	NOUN
ejpam-4854	45	15	-	-	PROPN
ejpam-4854	45	16	riordan	riordan	PROPN
ejpam-4854	45	17	maxtrix	maxtrix	PROPN
ejpam-4854	45	18	,	,	PUNCT
ejpam-4854	45	19	denoted	denote	VERB
ejpam-4854	45	20	by	by	ADP
ejpam-4854	45	21	⟨g	⟨g	NOUN
ejpam-4854	45	22	,	,	PUNCT
ejpam-4854	45	23	f⟩.	f⟩.	PROPN
ejpam-4854	45	24	for	for	ADP
ejpam-4854	45	25	example	example	NOUN
ejpam-4854	45	26	,	,	PUNCT
ejpam-4854	45	27	the	the	DET
ejpam-4854	45	28	e	e	PROPN
ejpam-4854	45	29	-	-	NOUN
ejpam-4854	45	30	riordan	riordan	PROPN
ejpam-4854	45	31	matrix	matrix	NOUN
ejpam-4854	45	32	representations	representation	NOUN
ejpam-4854	45	33	of	of	ADP
ejpam-4854	45	34	the	the	DET
ejpam-4854	45	35	three	three	NUM
ejpam-4854	45	36	common	common	ADJ
ejpam-4854	45	37	e	e	NOUN
ejpam-4854	45	38	-	-	NOUN
ejpam-4854	45	39	riordan	riordan	ADJ
ejpam-4854	45	40	matrices	matrix	NOUN
ejpam-4854	45	41	used	use	VERB
ejpam-4854	45	42	in	in	ADP
ejpam-4854	45	43	this	this	DET
ejpam-4854	45	44	paper	paper	NOUN
ejpam-4854	45	45	the	the	DET
ejpam-4854	45	46	pascal	pascal	ADJ
ejpam-4854	45	47	matrix	matrix	NOUN
ejpam-4854	45	48	,	,	PUNCT
ejpam-4854	45	49	stirling	stirling	NOUN
ejpam-4854	45	50	matrix	matrix	NOUN
ejpam-4854	45	51	of	of	ADP
ejpam-4854	45	52	the	the	DET
ejpam-4854	45	53	second	second	ADJ
ejpam-4854	45	54	kind	kind	NOUN
ejpam-4854	45	55	,	,	PUNCT
ejpam-4854	45	56	and	and	CCONJ
ejpam-4854	45	57	the	the	DET
ejpam-4854	45	58	(	(	PUNCT
ejpam-4854	45	59	r	r	NOUN
ejpam-4854	45	60	,	,	PUNCT
ejpam-4854	45	61	β)-stirling	β)-stirle	VERB
ejpam-4854	45	62	matrix	matrix	NOUN
ejpam-4854	45	63	:	:	PUNCT
ejpam-4854	45	64	pn	pn	PROPN
ejpam-4854	45	65	=	=	PROPN
ejpam-4854	45	66	〈	〈	PROPN
ejpam-4854	45	67	ez	ez	PROPN
ejpam-4854	45	68	,	,	PUNCT
ejpam-4854	45	69	z	z	NOUN
ejpam-4854	45	70	〉	〉	NOUN
ejpam-4854	45	71	,	,	PUNCT
ejpam-4854	45	72	s2(n	s2(n	PROPN
ejpam-4854	45	73	)	)	PUNCT
ejpam-4854	45	74	=	=	SYM
ejpam-4854	46	1	〈	〈	PROPN
ejpam-4854	46	2	1	1	NUM
ejpam-4854	46	3	,	,	PUNCT
ejpam-4854	46	4	ez	ez	PROPN
ejpam-4854	46	5	−	−	PROPN
ejpam-4854	46	6	1	1	NUM
ejpam-4854	46	7	〉	〉	NOUN
ejpam-4854	46	8	,	,	PUNCT
ejpam-4854	46	9	s(β	s(β	PROPN
ejpam-4854	46	10	,	,	PUNCT
ejpam-4854	46	11	r)(n	r)(n	PROPN
ejpam-4854	46	12	)	)	PUNCT
ejpam-4854	46	13	=	=	NUM
ejpam-4854	46	14	〈	〈	PROPN
ejpam-4854	46	15	e(r)z	e(r)z	NOUN
ejpam-4854	46	16	,	,	PUNCT
ejpam-4854	46	17	eβz	eβz	PROPN
ejpam-4854	46	18	−	−	PROPN
ejpam-4854	46	19	1	1	NUM
ejpam-4854	46	20	β	β	X
ejpam-4854	46	21	〉	〉	NOUN
ejpam-4854	46	22	.	.	PUNCT
ejpam-4854	47	1	g.	g.	PROPN
ejpam-4854	47	2	engalan	engalan	PROPN
ejpam-4854	47	3	,	,	PUNCT
ejpam-4854	47	4	m.r	m.r	PROPN
ejpam-4854	47	5	.	.	PROPN
ejpam-4854	47	6	latayada	latayada	PROPN
ejpam-4854	47	7	/	/	SYM
ejpam-4854	47	8	eur	eur	PROPN
ejpam-4854	47	9	.	.	PUNCT
ejpam-4854	48	1	j.	j.	PROPN
ejpam-4854	48	2	pure	pure	PROPN
ejpam-4854	48	3	appl	appl	PROPN
ejpam-4854	48	4	.	.	PROPN
ejpam-4854	48	5	math	math	PROPN
ejpam-4854	48	6	,	,	PUNCT
ejpam-4854	48	7	16	16	NUM
ejpam-4854	48	8	(	(	PUNCT
ejpam-4854	48	9	4	4	NUM
ejpam-4854	48	10	)	)	PUNCT
ejpam-4854	48	11	(	(	PUNCT
ejpam-4854	48	12	2023	2023	NUM
ejpam-4854	48	13	)	)	PUNCT
ejpam-4854	48	14	,	,	PUNCT
ejpam-4854	48	15	2306	2306	NUM
ejpam-4854	48	16	-	-	SYM
ejpam-4854	48	17	2322	2322	NUM
ejpam-4854	48	18	2308	2308	NUM
ejpam-4854	48	19	the	the	DET
ejpam-4854	48	20	set	set	NOUN
ejpam-4854	48	21	of	of	ADP
ejpam-4854	48	22	all	all	DET
ejpam-4854	48	23	e	e	NOUN
ejpam-4854	48	24	-	-	NOUN
ejpam-4854	48	25	riordan	riordan	ADJ
ejpam-4854	48	26	matrices	matrix	NOUN
ejpam-4854	48	27	forms	form	VERB
ejpam-4854	48	28	a	a	DET
ejpam-4854	48	29	group	group	NOUN
ejpam-4854	48	30	called	call	VERB
ejpam-4854	48	31	e	e	PROPN
ejpam-4854	48	32	-	-	PROPN
ejpam-4854	48	33	riordan	riordan	PROPN
ejpam-4854	48	34	group	group	NOUN
ejpam-4854	48	35	under	under	ADP
ejpam-4854	48	36	the	the	DET
ejpam-4854	48	37	riordan	riordan	PROPN
ejpam-4854	48	38	multiplication	multiplication	NOUN
ejpam-4854	48	39	defined	define	VERB
ejpam-4854	48	40	by	by	ADP
ejpam-4854	48	41	〈	〈	PROPN
ejpam-4854	48	42	g	g	NOUN
ejpam-4854	48	43	,	,	PUNCT
ejpam-4854	48	44	f	f	PROPN
ejpam-4854	48	45	〉	〉	PROPN
ejpam-4854	48	46	∗	∗	VERB
ejpam-4854	48	47	〈	〈	PROPN
ejpam-4854	48	48	h	h	NOUN
ejpam-4854	48	49	,	,	PUNCT
ejpam-4854	48	50	l	l	NOUN
ejpam-4854	48	51	〉	〉	NOUN
ejpam-4854	48	52	=	=	SYM
ejpam-4854	48	53	〈	〈	NOUN
ejpam-4854	48	54	gh(f	gh(f	NOUN
ejpam-4854	48	55	)	)	PUNCT
ejpam-4854	48	56	,	,	PUNCT
ejpam-4854	48	57	l(f	l(f	PROPN
ejpam-4854	48	58	)	)	PUNCT
ejpam-4854	48	59	〉	〉	NOUN
ejpam-4854	48	60	.	.	PUNCT
ejpam-4854	49	1	proposition	proposition	NOUN
ejpam-4854	49	2	1	1	NUM
ejpam-4854	49	3	.	.	PUNCT
ejpam-4854	50	1	let	let	VERB
ejpam-4854	50	2	pn	pn	PART
ejpam-4854	50	3	be	be	AUX
ejpam-4854	50	4	the	the	DET
ejpam-4854	50	5	n×	n×	PROPN
ejpam-4854	50	6	n	n	CCONJ
ejpam-4854	50	7	generalized	generalized	ADJ
ejpam-4854	50	8	pascal	pascal	ADJ
ejpam-4854	50	9	matrix	matrix	NOUN
ejpam-4854	50	10	,	,	PUNCT
ejpam-4854	50	11	then	then	ADV
ejpam-4854	50	12	s(β	s(β	PROPN
ejpam-4854	50	13	,	,	PUNCT
ejpam-4854	50	14	r)(n	r)(n	PROPN
ejpam-4854	50	15	)	)	PUNCT
ejpam-4854	50	16	=	=	SYM
ejpam-4854	50	17	pns	pns	NOUN
ejpam-4854	50	18	(	(	PUNCT
ejpam-4854	50	19	β	β	X
ejpam-4854	50	20	,	,	PUNCT
ejpam-4854	50	21	r−1)(n	r−1)(n	PROPN
ejpam-4854	50	22	)	)	PUNCT
ejpam-4854	50	23	proof	proof	NOUN
ejpam-4854	50	24	.	.	PUNCT
ejpam-4854	51	1	consider	consider	VERB
ejpam-4854	51	2	the	the	DET
ejpam-4854	51	3	e	e	NOUN
ejpam-4854	51	4	-	-	NOUN
ejpam-4854	51	5	riordan	riordan	PROPN
ejpam-4854	51	6	matrix	matrix	NOUN
ejpam-4854	51	7	representations	representation	NOUN
ejpam-4854	51	8	,	,	PUNCT
ejpam-4854	51	9	pn	pn	PROPN
ejpam-4854	51	10	=	=	PROPN
ejpam-4854	51	11	〈	〈	PROPN
ejpam-4854	51	12	ez	ez	PROPN
ejpam-4854	51	13	,	,	PUNCT
ejpam-4854	51	14	z	z	NOUN
ejpam-4854	51	15	〉	〉	NOUN
ejpam-4854	51	16	and	and	CCONJ
ejpam-4854	51	17	s(β	s(β	PROPN
ejpam-4854	51	18	,	,	PUNCT
ejpam-4854	51	19	r−1)(n	r−1)(n	PROPN
ejpam-4854	51	20	)	)	PUNCT
ejpam-4854	52	1	=	=	SYM
ejpam-4854	52	2	〈	〈	PROPN
ejpam-4854	52	3	e(r−1)z	e(r−1)z	PROPN
ejpam-4854	52	4	,	,	PUNCT
ejpam-4854	52	5	e	e	PROPN
ejpam-4854	52	6	βz−1	βz−1	PROPN
ejpam-4854	52	7	β	β	X
ejpam-4854	52	8	〉	〉	NOUN
ejpam-4854	52	9	.	.	PUNCT
ejpam-4854	53	1	by	by	ADP
ejpam-4854	53	2	using	use	VERB
ejpam-4854	53	3	the	the	DET
ejpam-4854	53	4	e	e	PROPN
ejpam-4854	53	5	-	-	PROPN
ejpam-4854	53	6	riordan	riordan	PROPN
ejpam-4854	53	7	matrix	matrix	NOUN
ejpam-4854	53	8	multiplication	multiplication	NOUN
ejpam-4854	53	9	,	,	PUNCT
ejpam-4854	53	10	we	we	PRON
ejpam-4854	53	11	have	have	VERB
ejpam-4854	53	12	pns	pns	NOUN
ejpam-4854	53	13	(	(	PUNCT
ejpam-4854	53	14	β	β	X
ejpam-4854	53	15	,	,	PUNCT
ejpam-4854	53	16	r−1)(n	r−1)(n	PROPN
ejpam-4854	53	17	)	)	PUNCT
ejpam-4854	53	18	=	=	PUNCT
ejpam-4854	53	19	〈	〈	PROPN
ejpam-4854	53	20	ez	ez	PROPN
ejpam-4854	53	21	,	,	PUNCT
ejpam-4854	53	22	z	z	PROPN
ejpam-4854	53	23	〉	〉	NOUN
ejpam-4854	53	24	∗	∗	X
ejpam-4854	53	25	〈	〈	PROPN
ejpam-4854	53	26	e(r−1)z	e(r−1)z	PROPN
ejpam-4854	53	27	,	,	PUNCT
ejpam-4854	53	28	eβz	eβz	PROPN
ejpam-4854	53	29	−	−	PROPN
ejpam-4854	53	30	1	1	NUM
ejpam-4854	53	31	β	β	X
ejpam-4854	53	32	〉	〉	NOUN
ejpam-4854	53	33	=	=	SYM
ejpam-4854	53	34	〈	〈	PROPN
ejpam-4854	53	35	eze(r−1)z	eze(r−1)z	NOUN
ejpam-4854	53	36	,	,	PUNCT
ejpam-4854	53	37	eβz	eβz	NOUN
ejpam-4854	53	38	−	−	PROPN
ejpam-4854	53	39	1	1	NUM
ejpam-4854	53	40	β	β	X
ejpam-4854	53	41	〉	〉	NOUN
ejpam-4854	53	42	=	=	SYM
ejpam-4854	53	43	〈	〈	PROPN
ejpam-4854	53	44	ez+(r−1)z	ez+(r−1)z	PROPN
ejpam-4854	53	45	,	,	PUNCT
ejpam-4854	53	46	eβz	eβz	NOUN
ejpam-4854	53	47	−	−	PROPN
ejpam-4854	53	48	1	1	NUM
ejpam-4854	53	49	β	β	X
ejpam-4854	53	50	〉	〉	NOUN
ejpam-4854	53	51	=	=	PUNCT
ejpam-4854	53	52	〈	〈	PROPN
ejpam-4854	53	53	ez+rz−z	ez+rz−z	NOUN
ejpam-4854	53	54	,	,	PUNCT
ejpam-4854	53	55	eβz	eβz	X
ejpam-4854	53	56	−	−	PROPN
ejpam-4854	53	57	1	1	NUM
ejpam-4854	53	58	β	β	X
ejpam-4854	53	59	〉	〉	NOUN
ejpam-4854	53	60	=	=	SYM
ejpam-4854	53	61	〈	〈	PROPN
ejpam-4854	53	62	erz	erz	PROPN
ejpam-4854	53	63	,	,	PUNCT
ejpam-4854	53	64	eβz	eβz	NOUN
ejpam-4854	53	65	−	−	PROPN
ejpam-4854	53	66	1	1	NUM
ejpam-4854	53	67	β	β	X
ejpam-4854	53	68	〉	〉	NOUN
ejpam-4854	53	69	=	=	SYM
ejpam-4854	53	70	s(β	s(β	PROPN
ejpam-4854	53	71	,	,	PUNCT
ejpam-4854	53	72	r)(n	r)(n	PROPN
ejpam-4854	53	73	)	)	PUNCT
ejpam-4854	53	74	.	.	PUNCT
ejpam-4854	54	1	example	example	NOUN
ejpam-4854	55	1	2	2	NUM
ejpam-4854	55	2	.	.	PUNCT
ejpam-4854	55	3	let	let	VERB
ejpam-4854	55	4	n	n	NOUN
ejpam-4854	55	5	=	=	SYM
ejpam-4854	55	6	4	4	X
ejpam-4854	55	7	.	.	PUNCT
ejpam-4854	56	1	then	then	ADV
ejpam-4854	56	2	p4s	p4s	PROPN
ejpam-4854	56	3	(	(	PUNCT
ejpam-4854	56	4	β	β	X
ejpam-4854	56	5	,	,	PUNCT
ejpam-4854	56	6	r−1)(4	r−1)(4	NOUN
ejpam-4854	56	7	)	)	PUNCT
ejpam-4854	56	8	=	=	SYM
ejpam-4854	56	9			NOUN
ejpam-4854	56	10	1	1	NUM
ejpam-4854	56	11	0	0	NUM
ejpam-4854	56	12	0	0	NUM
ejpam-4854	56	13	0	0	NUM
ejpam-4854	56	14	1	1	NUM
ejpam-4854	56	15	1	1	NUM
ejpam-4854	56	16	0	0	NUM
ejpam-4854	56	17	0	0	NUM
ejpam-4854	56	18	1	1	NUM
ejpam-4854	56	19	2	2	NUM
ejpam-4854	56	20	1	1	NUM
ejpam-4854	56	21	0	0	NUM
ejpam-4854	56	22	1	1	NUM
ejpam-4854	56	23	3	3	NUM
ejpam-4854	56	24	3	3	NUM
ejpam-4854	56	25	1	1	NUM
ejpam-4854	56	26			NOUN
ejpam-4854	56	27			NOUN
ejpam-4854	56	28	1	1	NUM
ejpam-4854	56	29	0	0	NUM
ejpam-4854	56	30	0	0	NUM
ejpam-4854	56	31	0	0	NUM
ejpam-4854	57	1	r	r	NOUN
ejpam-4854	57	2	−	−	NUM
ejpam-4854	57	3	1	1	NUM
ejpam-4854	57	4	1	1	NUM
ejpam-4854	57	5	0	0	NUM
ejpam-4854	57	6	0	0	NUM
ejpam-4854	57	7	(	(	PUNCT
ejpam-4854	57	8	r	r	NOUN
ejpam-4854	57	9	−	−	PROPN
ejpam-4854	57	10	1)2	1)2	NUM
ejpam-4854	57	11	β	β	X
ejpam-4854	58	1	+	+	CCONJ
ejpam-4854	59	1	2(r	2(r	NUM
ejpam-4854	59	2	−	−	NUM
ejpam-4854	59	3	1	1	NUM
ejpam-4854	59	4	)	)	PUNCT
ejpam-4854	59	5	1	1	NUM
ejpam-4854	59	6	0	0	NUM
ejpam-4854	60	1	(	(	PUNCT
ejpam-4854	60	2	r	r	NOUN
ejpam-4854	60	3	−	−	PROPN
ejpam-4854	60	4	1)3	1)3	PROPN
ejpam-4854	60	5	β2	β2	NOUN
ejpam-4854	60	6	+	+	CCONJ
ejpam-4854	60	7	3β(r	3β(r	NUM
ejpam-4854	60	8	−	−	NOUN
ejpam-4854	60	9	1	1	NUM
ejpam-4854	60	10	)	)	PUNCT
ejpam-4854	60	11	+	+	CCONJ
ejpam-4854	60	12	3(r	3(r	NUM
ejpam-4854	60	13	−	−	NUM
ejpam-4854	60	14	1)2	1)2	NUM
ejpam-4854	60	15	3β	3β	NOUN
ejpam-4854	61	1	+	+	CCONJ
ejpam-4854	61	2	3r	3r	NUM
ejpam-4854	61	3	1	1	NUM
ejpam-4854	61	4			NOUN
ejpam-4854	61	5	=	=	SYM
ejpam-4854	61	6			NOUN
ejpam-4854	61	7	1	1	NUM
ejpam-4854	61	8	0	0	NUM
ejpam-4854	61	9	0	0	NUM
ejpam-4854	61	10	0	0	NUM
ejpam-4854	62	1	r	r	NOUN
ejpam-4854	62	2	1	1	NUM
ejpam-4854	62	3	0	0	NUM
ejpam-4854	62	4	0	0	NUM
ejpam-4854	62	5	r2	r2	NOUN
ejpam-4854	62	6	β	β	X
ejpam-4854	62	7	+	+	CCONJ
ejpam-4854	62	8	2r	2r	NUM
ejpam-4854	62	9	1	1	NUM
ejpam-4854	62	10	0	0	NUM
ejpam-4854	62	11	r3	r3	PROPN
ejpam-4854	62	12	β2	β2	NOUN
ejpam-4854	62	13	+	+	CCONJ
ejpam-4854	62	14	3βr	3βr	ADJ
ejpam-4854	62	15	+	+	CCONJ
ejpam-4854	62	16	3r2	3r2	NUM
ejpam-4854	62	17	3β	3β	NUM
ejpam-4854	62	18	+	+	CCONJ
ejpam-4854	62	19	3r	3r	NUM
ejpam-4854	62	20	1	1	NUM
ejpam-4854	62	21			NOUN
ejpam-4854	62	22	=	=	SYM
ejpam-4854	62	23	s(β	s(β	PROPN
ejpam-4854	62	24	,	,	PUNCT
ejpam-4854	62	25	r)(4	r)(4	NUM
ejpam-4854	62	26	)	)	PUNCT
ejpam-4854	62	27	proposition	proposition	NOUN
ejpam-4854	62	28	2	2	NUM
ejpam-4854	62	29	.	.	PUNCT
ejpam-4854	62	30	let	let	VERB
ejpam-4854	62	31	pn[r	pn[r	PROPN
ejpam-4854	62	32	−	−	PROPN
ejpam-4854	62	33	s	s	PART
ejpam-4854	62	34	]	]	X
ejpam-4854	62	35	be	be	AUX
ejpam-4854	62	36	the	the	DET
ejpam-4854	62	37	n×	n×	PROPN
ejpam-4854	62	38	n	n	CCONJ
ejpam-4854	62	39	pascal	pascal	ADJ
ejpam-4854	62	40	matrix	matrix	NOUN
ejpam-4854	62	41	defined	define	VERB
ejpam-4854	62	42	by	by	ADP
ejpam-4854	62	43	pn[r	pn[r	PROPN
ejpam-4854	62	44	−	−	PROPN
ejpam-4854	62	45	s	s	NOUN
ejpam-4854	63	1	]	]	X
ejpam-4854	63	2	=	=	X
ejpam-4854	63	3	[	[	PUNCT
ejpam-4854	63	4	(	(	PUNCT
ejpam-4854	63	5	r	r	NOUN
ejpam-4854	63	6	−	−	PROPN
ejpam-4854	63	7	s)i−j	s)i−j	PROPN
ejpam-4854	63	8	(	(	PUNCT
ejpam-4854	63	9	n	n	X
ejpam-4854	63	10	k	k	PROPN
ejpam-4854	63	11	)	)	PUNCT
ejpam-4854	63	12	]	]	PUNCT
ejpam-4854	64	1	0≤i	0≤i	PROPN
ejpam-4854	64	2	,	,	PUNCT
ejpam-4854	64	3	j≤n−1	j≤n−1	PROPN
ejpam-4854	64	4	.	.	PUNCT
ejpam-4854	65	1	g.	g.	PROPN
ejpam-4854	65	2	engalan	engalan	PROPN
ejpam-4854	65	3	,	,	PUNCT
ejpam-4854	65	4	m.r	m.r	PROPN
ejpam-4854	65	5	.	.	PROPN
ejpam-4854	65	6	latayada	latayada	PROPN
ejpam-4854	65	7	/	/	SYM
ejpam-4854	65	8	eur	eur	PROPN
ejpam-4854	65	9	.	.	PUNCT
ejpam-4854	66	1	j.	j.	PROPN
ejpam-4854	66	2	pure	pure	PROPN
ejpam-4854	66	3	appl	appl	PROPN
ejpam-4854	66	4	.	.	PROPN
ejpam-4854	66	5	math	math	PROPN
ejpam-4854	66	6	,	,	PUNCT
ejpam-4854	66	7	16	16	NUM
ejpam-4854	66	8	(	(	PUNCT
ejpam-4854	66	9	4	4	NUM
ejpam-4854	66	10	)	)	PUNCT
ejpam-4854	66	11	(	(	PUNCT
ejpam-4854	66	12	2023	2023	NUM
ejpam-4854	66	13	)	)	PUNCT
ejpam-4854	66	14	,	,	PUNCT
ejpam-4854	66	15	2306	2306	NUM
ejpam-4854	66	16	-	-	SYM
ejpam-4854	66	17	2322	2322	NUM
ejpam-4854	66	18	2309	2309	NUM
ejpam-4854	66	19	then	then	ADV
ejpam-4854	66	20	,	,	PUNCT
ejpam-4854	66	21	s(β	s(β	PROPN
ejpam-4854	66	22	,	,	PUNCT
ejpam-4854	66	23	r)(n	r)(n	PROPN
ejpam-4854	66	24	)	)	PUNCT
ejpam-4854	66	25	=	=	SYM
ejpam-4854	66	26	pn[r	pn[r	PROPN
ejpam-4854	66	27	−	−	PROPN
ejpam-4854	66	28	s]s(β	s]s(β	NOUN
ejpam-4854	66	29	,	,	PUNCT
ejpam-4854	66	30	s)(n	s)(n	PROPN
ejpam-4854	66	31	)	)	PUNCT
ejpam-4854	66	32	,	,	PUNCT
ejpam-4854	66	33	provided	provide	VERB
ejpam-4854	66	34	that	that	SCONJ
ejpam-4854	66	35	r	r	NOUN
ejpam-4854	66	36	≥	≥	NOUN
ejpam-4854	66	37	s.	s.	PROPN
ejpam-4854	66	38	proof	proof	PROPN
ejpam-4854	66	39	.	.	PUNCT
ejpam-4854	67	1	consider	consider	VERB
ejpam-4854	67	2	pn[r	pn[r	NOUN
ejpam-4854	67	3	−	−	PROPN
ejpam-4854	67	4	s	s	X
ejpam-4854	67	5	]	]	X
ejpam-4854	67	6	=	=	PUNCT
ejpam-4854	67	7	⟨e(r−s)z	⟨e(r−s)z	X
ejpam-4854	67	8	,	,	PUNCT
ejpam-4854	67	9	z⟩	z⟩	NOUN
ejpam-4854	67	10	and	and	CCONJ
ejpam-4854	67	11	s(β	s(β	PROPN
ejpam-4854	67	12	,	,	PUNCT
ejpam-4854	67	13	s)(n	s)(n	PROPN
ejpam-4854	67	14	)	)	PUNCT
ejpam-4854	68	1	=	=	SYM
ejpam-4854	68	2	⟨erz	⟨erz	PROPN
ejpam-4854	68	3	,	,	PUNCT
ejpam-4854	68	4	eβz−1	eβz−1	PROPN
ejpam-4854	68	5	β	β	PROPN
ejpam-4854	68	6	⟩.	⟩.	PROPN
ejpam-4854	68	7	then	then	ADV
ejpam-4854	68	8	,	,	PUNCT
ejpam-4854	68	9	pn[r	pn[r	PROPN
ejpam-4854	68	10	−	−	PROPN
ejpam-4854	68	11	s]s(β	s]s(β	NOUN
ejpam-4854	68	12	,	,	PUNCT
ejpam-4854	68	13	s)(n	s)(n	PROPN
ejpam-4854	68	14	)	)	PUNCT
ejpam-4854	68	15	=	=	SYM
ejpam-4854	69	1	〈	〈	PROPN
ejpam-4854	69	2	e(r−s)z	e(r−s)z	NOUN
ejpam-4854	69	3	,	,	PUNCT
ejpam-4854	69	4	z	z	NOUN
ejpam-4854	69	5	〉	〉	NOUN
ejpam-4854	69	6	∗	∗	VERB
ejpam-4854	69	7	〈	〈	PROPN
ejpam-4854	69	8	esz	esz	NOUN
ejpam-4854	69	9	,	,	PUNCT
ejpam-4854	69	10	eβz	eβz	NOUN
ejpam-4854	69	11	−	−	PROPN
ejpam-4854	69	12	1	1	NUM
ejpam-4854	69	13	β	β	X
ejpam-4854	69	14	〉	〉	NOUN
ejpam-4854	69	15	=	=	PUNCT
ejpam-4854	69	16	〈	〈	PROPN
ejpam-4854	69	17	e(r−s)zesz	e(r−s)zesz	PROPN
ejpam-4854	69	18	,	,	PUNCT
ejpam-4854	69	19	eβz	eβz	NOUN
ejpam-4854	69	20	−	−	PROPN
ejpam-4854	69	21	1	1	NUM
ejpam-4854	69	22	β	β	X
ejpam-4854	69	23	〉	〉	NOUN
ejpam-4854	69	24	=	=	SYM
ejpam-4854	69	25	〈	〈	PROPN
ejpam-4854	69	26	e(r−s)z+sz	e(r−s)z+sz	PROPN
ejpam-4854	69	27	,	,	PUNCT
ejpam-4854	69	28	eβz	eβz	PROPN
ejpam-4854	69	29	−	−	PROPN
ejpam-4854	69	30	1	1	NUM
ejpam-4854	69	31	β	β	X
ejpam-4854	69	32	〉	〉	NOUN
ejpam-4854	69	33	=	=	SYM
ejpam-4854	69	34	〈	〈	PROPN
ejpam-4854	69	35	erz−sz+sz	erz−sz+sz	NOUN
ejpam-4854	69	36	,	,	PUNCT
ejpam-4854	69	37	eβz	eβz	NOUN
ejpam-4854	69	38	−	−	PROPN
ejpam-4854	69	39	1	1	NUM
ejpam-4854	69	40	β	β	X
ejpam-4854	69	41	〉	〉	NOUN
ejpam-4854	69	42	=	=	SYM
ejpam-4854	69	43	〈	〈	PROPN
ejpam-4854	69	44	erz	erz	PROPN
ejpam-4854	69	45	,	,	PUNCT
ejpam-4854	69	46	eβz	eβz	NOUN
ejpam-4854	69	47	−	−	PROPN
ejpam-4854	69	48	1	1	NUM
ejpam-4854	69	49	β	β	X
ejpam-4854	69	50	〉	〉	NOUN
ejpam-4854	69	51	=	=	SYM
ejpam-4854	69	52	s(β	s(β	PROPN
ejpam-4854	69	53	,	,	PUNCT
ejpam-4854	69	54	r)(n	r)(n	PROPN
ejpam-4854	69	55	)	)	PUNCT
ejpam-4854	69	56	.	.	PUNCT
ejpam-4854	70	1	example	example	NOUN
ejpam-4854	71	1	3	3	X
ejpam-4854	71	2	.	.	PUNCT
ejpam-4854	71	3	let	let	VERB
ejpam-4854	71	4	n	n	NOUN
ejpam-4854	71	5	=	=	SYM
ejpam-4854	71	6	4	4	X
ejpam-4854	71	7	.	.	PUNCT
ejpam-4854	71	8	then	then	ADV
ejpam-4854	71	9	p4[r	p4[r	NOUN
ejpam-4854	71	10	−	−	PROPN
ejpam-4854	71	11	s]s(β	s]s(β	NOUN
ejpam-4854	71	12	,	,	PUNCT
ejpam-4854	71	13	s)(4	s)(4	NUM
ejpam-4854	71	14	)	)	PUNCT
ejpam-4854	71	15	=	=	SYM
ejpam-4854	71	16			NOUN
ejpam-4854	72	1	1	1	NUM
ejpam-4854	72	2	0	0	NUM
ejpam-4854	72	3	0	0	NUM
ejpam-4854	72	4	0	0	NUM
ejpam-4854	73	1	r	r	NOUN
ejpam-4854	74	1	−	−	NOUN
ejpam-4854	74	2	s	s	NOUN
ejpam-4854	74	3	1	1	NUM
ejpam-4854	74	4	0	0	NUM
ejpam-4854	74	5	0	0	NUM
ejpam-4854	75	1	(	(	PUNCT
ejpam-4854	75	2	r	r	NOUN
ejpam-4854	75	3	−	−	NOUN
ejpam-4854	75	4	s)2	s)2	NOUN
ejpam-4854	75	5	2(r	2(r	NUM
ejpam-4854	75	6	−	−	NUM
ejpam-4854	75	7	s	s	X
ejpam-4854	75	8	)	)	PUNCT
ejpam-4854	75	9	1	1	NUM
ejpam-4854	75	10	0	0	NUM
ejpam-4854	76	1	(	(	PUNCT
ejpam-4854	76	2	r	r	NOUN
ejpam-4854	76	3	−	−	PROPN
ejpam-4854	76	4	s)3	s)3	ADJ
ejpam-4854	76	5	3(r	3(r	NUM
ejpam-4854	76	6	−	−	NOUN
ejpam-4854	76	7	s)2	s)2	NOUN
ejpam-4854	76	8	3(r	3(r	NUM
ejpam-4854	76	9	−	−	PROPN
ejpam-4854	76	10	s	s	PART
ejpam-4854	76	11	)	)	PUNCT
ejpam-4854	76	12	1	1	NUM
ejpam-4854	76	13			NOUN
ejpam-4854	76	14			NOUN
ejpam-4854	76	15	1	1	NUM
ejpam-4854	76	16	0	0	NUM
ejpam-4854	76	17	0	0	NUM
ejpam-4854	76	18	0	0	NUM
ejpam-4854	76	19	s	s	PART
ejpam-4854	76	20	1	1	NUM
ejpam-4854	76	21	0	0	NUM
ejpam-4854	76	22	0	0	NUM
ejpam-4854	76	23	s2	s2	NOUN
ejpam-4854	76	24	β	β	X
ejpam-4854	77	1	+	+	CCONJ
ejpam-4854	77	2	2s	2s	NUM
ejpam-4854	77	3	1	1	NUM
ejpam-4854	77	4	0	0	NUM
ejpam-4854	77	5	s3	s3	PROPN
ejpam-4854	77	6	β2	β2	NOUN
ejpam-4854	77	7	+	+	CCONJ
ejpam-4854	77	8	3βs+	3βs+	NUM
ejpam-4854	77	9	3s2	3s2	NUM
ejpam-4854	77	10	3β	3β	NUM
ejpam-4854	78	1	+	+	CCONJ
ejpam-4854	78	2	3s	3s	NUM
ejpam-4854	78	3	1	1	NUM
ejpam-4854	78	4			NOUN
ejpam-4854	78	5	=	=	PUNCT
ejpam-4854	78	6			NOUN
ejpam-4854	78	7	1	1	NUM
ejpam-4854	78	8	0	0	NUM
ejpam-4854	78	9	0	0	NUM
ejpam-4854	78	10	0	0	NUM
ejpam-4854	79	1	r	r	NOUN
ejpam-4854	79	2	1	1	NUM
ejpam-4854	79	3	0	0	NUM
ejpam-4854	79	4	0	0	NUM
ejpam-4854	79	5	−1	−1	NOUN
ejpam-4854	80	1	+	+	SYM
ejpam-4854	80	2	2r	2r	NUM
ejpam-4854	81	1	+	+	CCONJ
ejpam-4854	81	2	(	(	PUNCT
ejpam-4854	81	3	r	r	NOUN
ejpam-4854	81	4	−	−	PROPN
ejpam-4854	81	5	1)2	1)2	NUM
ejpam-4854	81	6	b+	b+	PUNCT
ejpam-4854	81	7	2r	2r	NUM
ejpam-4854	81	8	1	1	NUM
ejpam-4854	81	9	0	0	NUM
ejpam-4854	81	10	−2	−2	NOUN
ejpam-4854	81	11	+	+	CCONJ
ejpam-4854	81	12	3r	3r	NUM
ejpam-4854	81	13	+	+	SYM
ejpam-4854	81	14	3	3	NUM
ejpam-4854	81	15	(	(	PUNCT
ejpam-4854	81	16	r	r	NOUN
ejpam-4854	81	17	−	−	PROPN
ejpam-4854	81	18	1)2	1)2	NUM
ejpam-4854	82	1	+	+	CCONJ
ejpam-4854	82	2	(	(	PUNCT
ejpam-4854	82	3	r	r	NOUN
ejpam-4854	82	4	−	−	PROPN
ejpam-4854	82	5	1)3	1)3	PROPN
ejpam-4854	82	6	b2	b2	NOUN
ejpam-4854	82	7	+	+	CCONJ
ejpam-4854	82	8	3b+	3b+	NUM
ejpam-4854	82	9	6r	6r	NUM
ejpam-4854	82	10	−	−	NUM
ejpam-4854	82	11	3	3	NUM
ejpam-4854	82	12	+	+	NUM
ejpam-4854	82	13	3b	3b	NUM
ejpam-4854	82	14	(	(	PUNCT
ejpam-4854	82	15	r	r	NOUN
ejpam-4854	82	16	−	−	PROPN
ejpam-4854	82	17	1	1	NUM
ejpam-4854	82	18	)	)	PUNCT
ejpam-4854	82	19	+	+	CCONJ
ejpam-4854	82	20	3	3	NUM
ejpam-4854	82	21	(	(	PUNCT
ejpam-4854	82	22	r	r	NOUN
ejpam-4854	82	23	−	−	PROPN
ejpam-4854	82	24	1)2	1)2	NUM
ejpam-4854	82	25	3b+	3b+	NUM
ejpam-4854	82	26	3r	3r	NUM
ejpam-4854	82	27	1	1	NUM
ejpam-4854	82	28			NOUN
ejpam-4854	82	29	=	=	PUNCT
ejpam-4854	82	30			NOUN
ejpam-4854	82	31	1	1	NUM
ejpam-4854	82	32	0	0	NUM
ejpam-4854	82	33	0	0	NUM
ejpam-4854	82	34	0	0	NUM
ejpam-4854	83	1	r	r	NOUN
ejpam-4854	83	2	1	1	NUM
ejpam-4854	83	3	0	0	NUM
ejpam-4854	83	4	0	0	NUM
ejpam-4854	83	5	r2	r2	NOUN
ejpam-4854	83	6	β	β	X
ejpam-4854	83	7	+	+	CCONJ
ejpam-4854	83	8	2r	2r	NUM
ejpam-4854	83	9	1	1	NUM
ejpam-4854	83	10	0	0	NUM
ejpam-4854	83	11	r3	r3	PROPN
ejpam-4854	83	12	β2	β2	NOUN
ejpam-4854	83	13	+	+	CCONJ
ejpam-4854	83	14	3βr	3βr	ADJ
ejpam-4854	83	15	+	+	CCONJ
ejpam-4854	83	16	3r2	3r2	NUM
ejpam-4854	83	17	3β	3β	NUM
ejpam-4854	83	18	+	+	CCONJ
ejpam-4854	83	19	3r	3r	NUM
ejpam-4854	83	20	1	1	NUM
ejpam-4854	83	21			NOUN
ejpam-4854	83	22	=	=	SYM
ejpam-4854	83	23	s(β	s(β	PROPN
ejpam-4854	83	24	,	,	PUNCT
ejpam-4854	83	25	r)(4	r)(4	NUM
ejpam-4854	83	26	)	)	PUNCT
ejpam-4854	83	27	proposition	proposition	NOUN
ejpam-4854	83	28	3	3	X
ejpam-4854	83	29	.	.	PUNCT
ejpam-4854	83	30	let	let	VERB
ejpam-4854	83	31	pn[r	pn[r	PROPN
ejpam-4854	83	32	]	]	PUNCT
ejpam-4854	83	33	and	and	CCONJ
ejpam-4854	83	34	s2(n)[β	s2(n)[β	PROPN
ejpam-4854	83	35	]	]	PUNCT
ejpam-4854	83	36	be	be	VERB
ejpam-4854	83	37	the	the	DET
ejpam-4854	83	38	n×n	n×n	PROPN
ejpam-4854	83	39	pascal	pascal	ADJ
ejpam-4854	83	40	matrix	matrix	NOUN
ejpam-4854	83	41	and	and	CCONJ
ejpam-4854	83	42	stirling	stirling	NOUN
ejpam-4854	83	43	matrix	matrix	NOUN
ejpam-4854	83	44	of	of	ADP
ejpam-4854	83	45	the	the	DET
ejpam-4854	83	46	second	second	ADJ
ejpam-4854	83	47	kind	kind	NOUN
ejpam-4854	83	48	.	.	PUNCT
ejpam-4854	84	1	then	then	ADV
ejpam-4854	84	2	,	,	PUNCT
ejpam-4854	84	3	s(β	s(β	PROPN
ejpam-4854	84	4	,	,	PUNCT
ejpam-4854	84	5	r)(n	r)(n	PROPN
ejpam-4854	84	6	)	)	PUNCT
ejpam-4854	84	7	=	=	SYM
ejpam-4854	84	8	pn[r]s2(n)[β	pn[r]s2(n)[β	PROPN
ejpam-4854	84	9	]	]	PUNCT
ejpam-4854	84	10	,	,	PUNCT
ejpam-4854	84	11	where	where	SCONJ
ejpam-4854	84	12	(	(	PUNCT
ejpam-4854	84	13	s2(n)[β]i	s2(n)[β]i	PROPN
ejpam-4854	84	14	,	,	PUNCT
ejpam-4854	84	15	j	j	PROPN
ejpam-4854	84	16	=	=	SYM
ejpam-4854	84	17	βi−js(i	βi−js(i	PROPN
ejpam-4854	84	18	,	,	PUNCT
ejpam-4854	84	19	j	j	NOUN
ejpam-4854	84	20	)	)	PUNCT
ejpam-4854	84	21	and	and	CCONJ
ejpam-4854	84	22	s(i	s(i	PROPN
ejpam-4854	84	23	,	,	PUNCT
ejpam-4854	84	24	j	j	PROPN
ejpam-4854	84	25	)	)	PUNCT
ejpam-4854	84	26	is	be	AUX
ejpam-4854	84	27	a	a	DET
ejpam-4854	84	28	stirling	stirling	NOUN
ejpam-4854	84	29	number	number	NOUN
ejpam-4854	84	30	of	of	ADP
ejpam-4854	84	31	the	the	DET
ejpam-4854	84	32	second	second	ADJ
ejpam-4854	84	33	kind	kind	NOUN
ejpam-4854	84	34	and	and	CCONJ
ejpam-4854	85	1	0	0	NUM
ejpam-4854	85	2	≤	≤	NOUN
ejpam-4854	85	3	i	i	PRON
ejpam-4854	85	4	,	,	PUNCT
ejpam-4854	85	5	j	j	PROPN
ejpam-4854	85	6	≤	≤	PROPN
ejpam-4854	85	7	n−	n−	PROPN
ejpam-4854	85	8	1	1	NUM
ejpam-4854	85	9	.	.	PUNCT
ejpam-4854	85	10	g.	g.	PROPN
ejpam-4854	85	11	engalan	engalan	PROPN
ejpam-4854	85	12	,	,	PUNCT
ejpam-4854	85	13	m.r	m.r	PROPN
ejpam-4854	85	14	.	.	PROPN
ejpam-4854	85	15	latayada	latayada	PROPN
ejpam-4854	85	16	/	/	SYM
ejpam-4854	85	17	eur	eur	PROPN
ejpam-4854	85	18	.	.	PUNCT
ejpam-4854	86	1	j.	j.	PROPN
ejpam-4854	86	2	pure	pure	PROPN
ejpam-4854	86	3	appl	appl	PROPN
ejpam-4854	86	4	.	.	PROPN
ejpam-4854	86	5	math	math	PROPN
ejpam-4854	86	6	,	,	PUNCT
ejpam-4854	86	7	16	16	NUM
ejpam-4854	86	8	(	(	PUNCT
ejpam-4854	86	9	4	4	NUM
ejpam-4854	86	10	)	)	PUNCT
ejpam-4854	86	11	(	(	PUNCT
ejpam-4854	86	12	2023	2023	NUM
ejpam-4854	86	13	)	)	PUNCT
ejpam-4854	86	14	,	,	PUNCT
ejpam-4854	86	15	2306	2306	NUM
ejpam-4854	86	16	-	-	SYM
ejpam-4854	86	17	2322	2322	NUM
ejpam-4854	86	18	2310	2310	NUM
ejpam-4854	86	19	proof	proof	NOUN
ejpam-4854	86	20	.	.	PUNCT
ejpam-4854	87	1	it	it	PRON
ejpam-4854	87	2	was	be	AUX
ejpam-4854	87	3	previously	previously	ADV
ejpam-4854	87	4	shown	show	VERB
ejpam-4854	87	5	that	that	SCONJ
ejpam-4854	87	6	the	the	DET
ejpam-4854	87	7	e	e	PROPN
ejpam-4854	87	8	-	-	NOUN
ejpam-4854	87	9	riordan	riordan	PROPN
ejpam-4854	87	10	matrix	matrix	NOUN
ejpam-4854	87	11	representations	representation	NOUN
ejpam-4854	87	12	of	of	ADP
ejpam-4854	87	13	pn[r	pn[r	NOUN
ejpam-4854	87	14	]	]	PUNCT
ejpam-4854	87	15	and	and	CCONJ
ejpam-4854	87	16	s2(n)[β	s2(n)[β	PROPN
ejpam-4854	87	17	]	]	PUNCT
ejpam-4854	87	18	are	be	AUX
ejpam-4854	87	19	⟨erz	⟨erz	PROPN
ejpam-4854	87	20	,	,	PUNCT
ejpam-4854	87	21	z⟩	z⟩	PROPN
ejpam-4854	87	22	and	and	CCONJ
ejpam-4854	87	23	⟨1	⟨1	PROPN
ejpam-4854	87	24	,	,	PUNCT
ejpam-4854	87	25	eβz	eβz	NOUN
ejpam-4854	87	26	−	−	NOUN
ejpam-4854	87	27	1⟩	1⟩	NUM
ejpam-4854	87	28	,	,	PUNCT
ejpam-4854	87	29	respectively	respectively	ADV
ejpam-4854	87	30	.	.	PUNCT
ejpam-4854	88	1	using	use	VERB
ejpam-4854	88	2	the	the	DET
ejpam-4854	88	3	e	e	NOUN
ejpam-4854	88	4	-	-	NOUN
ejpam-4854	88	5	riordan	riordan	PROPN
ejpam-4854	88	6	multiplication	multiplication	NOUN
ejpam-4854	88	7	,	,	PUNCT
ejpam-4854	88	8	we	we	PRON
ejpam-4854	88	9	have	have	AUX
ejpam-4854	88	10	pn[r]s2(n)[β	pn[r]s2(n)[β	VERB
ejpam-4854	88	11	]	]	PUNCT
ejpam-4854	88	12	=	=	PUNCT
ejpam-4854	88	13	〈	〈	PROPN
ejpam-4854	88	14	erz	erz	NOUN
ejpam-4854	88	15	,	,	PUNCT
ejpam-4854	88	16	z	z	NOUN
ejpam-4854	88	17	〉	〉	NOUN
ejpam-4854	88	18	∗	∗	X
ejpam-4854	88	19	〈	〈	PROPN
ejpam-4854	88	20	1	1	NUM
ejpam-4854	88	21	,	,	PUNCT
ejpam-4854	88	22	eβz	eβz	X
ejpam-4854	88	23	−	−	PROPN
ejpam-4854	88	24	1	1	NUM
ejpam-4854	88	25	β	β	X
ejpam-4854	88	26	〉	〉	NOUN
ejpam-4854	88	27	=	=	PUNCT
ejpam-4854	88	28	〈	〈	NOUN
ejpam-4854	88	29	erz(1	erz(1	NOUN
ejpam-4854	88	30	)	)	PUNCT
ejpam-4854	88	31	,	,	PUNCT
ejpam-4854	88	32	eβz	eβz	PROPN
ejpam-4854	88	33	−	−	PROPN
ejpam-4854	88	34	1	1	NUM
ejpam-4854	88	35	β	β	X
ejpam-4854	88	36	〉	〉	NOUN
ejpam-4854	88	37	=	=	SYM
ejpam-4854	88	38	〈	〈	PROPN
ejpam-4854	88	39	erz	erz	PROPN
ejpam-4854	88	40	,	,	PUNCT
ejpam-4854	88	41	eβz	eβz	NOUN
ejpam-4854	88	42	−	−	PROPN
ejpam-4854	88	43	1	1	NUM
ejpam-4854	88	44	β	β	X
ejpam-4854	88	45	〉	〉	NOUN
ejpam-4854	88	46	=	=	SYM
ejpam-4854	88	47	s(β	s(β	PROPN
ejpam-4854	88	48	,	,	PUNCT
ejpam-4854	88	49	r)(n	r)(n	PROPN
ejpam-4854	88	50	)	)	PUNCT
ejpam-4854	88	51	.	.	PUNCT
ejpam-4854	89	1	example	example	NOUN
ejpam-4854	90	1	4	4	X
ejpam-4854	90	2	.	.	PUNCT
ejpam-4854	90	3	let	let	VERB
ejpam-4854	90	4	n	n	NOUN
ejpam-4854	90	5	=	=	SYM
ejpam-4854	90	6	4	4	X
ejpam-4854	90	7	.	.	PUNCT
ejpam-4854	90	8	then	then	ADV
ejpam-4854	90	9	p4[r]s2(4)[β	p4[r]s2(4)[β	VERB
ejpam-4854	90	10	]	]	X
ejpam-4854	90	11	=	=	SYM
ejpam-4854	90	12			NOUN
ejpam-4854	90	13	1	1	NUM
ejpam-4854	90	14	0	0	NUM
ejpam-4854	90	15	0	0	NUM
ejpam-4854	90	16	0	0	NUM
ejpam-4854	91	1	r	r	NOUN
ejpam-4854	91	2	1	1	NUM
ejpam-4854	91	3	0	0	NUM
ejpam-4854	91	4	0	0	NUM
ejpam-4854	91	5	r2	r2	NOUN
ejpam-4854	91	6	2r	2r	NUM
ejpam-4854	91	7	1	1	NUM
ejpam-4854	91	8	0	0	NUM
ejpam-4854	91	9	r3	r3	PROPN
ejpam-4854	91	10	3r2	3r2	NUM
ejpam-4854	91	11	3r	3r	NUM
ejpam-4854	91	12	1	1	NUM
ejpam-4854	91	13			NOUN
ejpam-4854	91	14			NOUN
ejpam-4854	91	15	1	1	NUM
ejpam-4854	91	16	0	0	NUM
ejpam-4854	91	17	0	0	NUM
ejpam-4854	91	18	0	0	NUM
ejpam-4854	91	19	0	0	NUM
ejpam-4854	91	20	1	1	NUM
ejpam-4854	91	21	0	0	NUM
ejpam-4854	91	22	0	0	NUM
ejpam-4854	91	23	0	0	NUM
ejpam-4854	91	24	β	β	NOUN
ejpam-4854	91	25	1	1	NUM
ejpam-4854	91	26	0	0	NUM
ejpam-4854	91	27	0	0	NUM
ejpam-4854	91	28	β2	β2	NOUN
ejpam-4854	91	29	3β	3β	NOUN
ejpam-4854	91	30	1	1	NUM
ejpam-4854	91	31			NOUN
ejpam-4854	91	32	=	=	SYM
ejpam-4854	91	33			NOUN
ejpam-4854	91	34	1	1	NUM
ejpam-4854	91	35	0	0	NUM
ejpam-4854	91	36	0	0	NUM
ejpam-4854	91	37	0	0	NUM
ejpam-4854	92	1	r	r	NOUN
ejpam-4854	92	2	1	1	NUM
ejpam-4854	92	3	0	0	NUM
ejpam-4854	92	4	0	0	NUM
ejpam-4854	92	5	r2	r2	NOUN
ejpam-4854	92	6	β	β	X
ejpam-4854	92	7	+	+	CCONJ
ejpam-4854	92	8	2r	2r	NUM
ejpam-4854	92	9	1	1	NUM
ejpam-4854	92	10	0	0	NUM
ejpam-4854	92	11	r3	r3	PROPN
ejpam-4854	92	12	β2	β2	NOUN
ejpam-4854	92	13	+	+	CCONJ
ejpam-4854	92	14	3βr	3βr	ADJ
ejpam-4854	92	15	+	+	CCONJ
ejpam-4854	92	16	3r2	3r2	NUM
ejpam-4854	92	17	3β	3β	NUM
ejpam-4854	92	18	+	+	CCONJ
ejpam-4854	92	19	3r	3r	NUM
ejpam-4854	92	20	1	1	NUM
ejpam-4854	92	21			NOUN
ejpam-4854	92	22	=	=	SYM
ejpam-4854	92	23	s(β	s(β	PROPN
ejpam-4854	92	24	,	,	PUNCT
ejpam-4854	92	25	r)(4	r)(4	NUM
ejpam-4854	92	26	)	)	PUNCT
ejpam-4854	92	27	proposition	proposition	NOUN
ejpam-4854	92	28	4	4	NUM
ejpam-4854	92	29	.	.	PUNCT
ejpam-4854	92	30	letpn[r	letpn[r	PROPN
ejpam-4854	93	1	−	−	PROPN
ejpam-4854	93	2	rβ	rβ	X
ejpam-4854	93	3	]	]	PUNCT
ejpam-4854	93	4	and	and	CCONJ
ejpam-4854	93	5	s(1,r)(n)[β	s(1,r)(n)[β	X
ejpam-4854	93	6	]	]	PUNCT
ejpam-4854	93	7	be	be	VERB
ejpam-4854	93	8	n×	n×	PROPN
ejpam-4854	93	9	n	n	PRON
ejpam-4854	93	10	matrices	matrix	NOUN
ejpam-4854	93	11	,	,	PUNCT
ejpam-4854	93	12	where	where	SCONJ
ejpam-4854	93	13	(	(	PUNCT
ejpam-4854	93	14	s(1,r)(n)[β])ij	s(1,r)(n)[β])ij	NOUN
ejpam-4854	93	15	=	=	SYM
ejpam-4854	93	16	βi−j	βi−j	PROPN
ejpam-4854	93	17	〈	〈	PROPN
ejpam-4854	93	18	n	n	CCONJ
ejpam-4854	93	19	k	k	X
ejpam-4854	93	20	〉	〉	NOUN
ejpam-4854	93	21	1,r	1,r	NUM
ejpam-4854	93	22	.	.	PUNCT
ejpam-4854	94	1	then	then	ADV
ejpam-4854	94	2	,	,	PUNCT
ejpam-4854	94	3	s(β	s(β	PROPN
ejpam-4854	94	4	,	,	PUNCT
ejpam-4854	94	5	r)(n	r)(n	PROPN
ejpam-4854	94	6	)	)	PUNCT
ejpam-4854	94	7	=	=	SYM
ejpam-4854	94	8	pn[r	pn[r	NOUN
ejpam-4854	94	9	−	−	PROPN
ejpam-4854	94	10	rβ]s(1,r)(n)[β	rβ]s(1,r)(n)[β	NOUN
ejpam-4854	94	11	]	]	PUNCT
ejpam-4854	94	12	proof	proof	NOUN
ejpam-4854	94	13	.	.	PUNCT
ejpam-4854	95	1	consider	consider	VERB
ejpam-4854	95	2	the	the	DET
ejpam-4854	95	3	e	e	NOUN
ejpam-4854	95	4	-	-	NOUN
ejpam-4854	95	5	riordan	riordan	PROPN
ejpam-4854	95	6	matrix	matrix	NOUN
ejpam-4854	95	7	representations	representation	VERB
ejpam-4854	95	8	pn[r	pn[r	PROPN
ejpam-4854	95	9	−	−	PROPN
ejpam-4854	95	10	rβ	rβ	X
ejpam-4854	95	11	]	]	X
ejpam-4854	95	12	=	=	PUNCT
ejpam-4854	95	13	〈	〈	NOUN
ejpam-4854	95	14	e(r−rβ)z	e(r−rβ)z	NOUN
ejpam-4854	95	15	,	,	PUNCT
ejpam-4854	95	16	z	z	NOUN
ejpam-4854	95	17	〉	〉	NOUN
ejpam-4854	95	18	and	and	CCONJ
ejpam-4854	95	19	s(1,r)(n)[β	s(1,r)(n)[β	NOUN
ejpam-4854	95	20	]	]	PUNCT
ejpam-4854	95	21	=	=	PUNCT
ejpam-4854	96	1	[	[	X
ejpam-4854	96	2	〈	〈	PROPN
ejpam-4854	96	3	erz	erz	PROPN
ejpam-4854	96	4	,	,	PUNCT
ejpam-4854	96	5	ez	ez	X
ejpam-4854	96	6	−	−	PROPN
ejpam-4854	96	7	1	1	NUM
ejpam-4854	96	8	1	1	NUM
ejpam-4854	96	9	〉	〉	NOUN
ejpam-4854	96	10	]	]	X
ejpam-4854	96	11	β	β	X
ejpam-4854	96	12	=	=	PUNCT
ejpam-4854	96	13	〈	〈	PROPN
ejpam-4854	96	14	erβz	erβz	NOUN
ejpam-4854	96	15	,	,	PUNCT
ejpam-4854	96	16	eβz	eβz	NOUN
ejpam-4854	96	17	−	−	PROPN
ejpam-4854	96	18	1	1	NUM
ejpam-4854	96	19	β	β	X
ejpam-4854	96	20	〉	〉	NOUN
ejpam-4854	96	21	g.	g.	PROPN
ejpam-4854	96	22	engalan	engalan	PROPN
ejpam-4854	96	23	,	,	PUNCT
ejpam-4854	96	24	m.r	m.r	PROPN
ejpam-4854	96	25	.	.	PROPN
ejpam-4854	96	26	latayada	latayada	PROPN
ejpam-4854	96	27	/	/	SYM
ejpam-4854	96	28	eur	eur	PROPN
ejpam-4854	96	29	.	.	PUNCT
ejpam-4854	97	1	j.	j.	PROPN
ejpam-4854	97	2	pure	pure	PROPN
ejpam-4854	97	3	appl	appl	PROPN
ejpam-4854	97	4	.	.	PROPN
ejpam-4854	97	5	math	math	PROPN
ejpam-4854	97	6	,	,	PUNCT
ejpam-4854	97	7	16	16	NUM
ejpam-4854	97	8	(	(	PUNCT
ejpam-4854	97	9	4	4	NUM
ejpam-4854	97	10	)	)	PUNCT
ejpam-4854	97	11	(	(	PUNCT
ejpam-4854	97	12	2023	2023	NUM
ejpam-4854	97	13	)	)	PUNCT
ejpam-4854	97	14	,	,	PUNCT
ejpam-4854	97	15	2306	2306	NUM
ejpam-4854	97	16	-	-	SYM
ejpam-4854	97	17	2322	2322	NUM
ejpam-4854	97	18	2311	2311	NUM
ejpam-4854	97	19	now	now	ADV
ejpam-4854	97	20	,	,	PUNCT
ejpam-4854	97	21	pn[r	pn[r	NOUN
ejpam-4854	97	22	−	−	PROPN
ejpam-4854	97	23	rβ]s(1,r)(n)[β	rβ]s(1,r)(n)[β	NOUN
ejpam-4854	97	24	]	]	X
ejpam-4854	97	25	=	=	PUNCT
ejpam-4854	97	26	〈	〈	NOUN
ejpam-4854	97	27	e(r−rβ)z	e(r−rβ)z	NOUN
ejpam-4854	97	28	,	,	PUNCT
ejpam-4854	97	29	z	z	NOUN
ejpam-4854	97	30	〉	〉	NOUN
ejpam-4854	97	31	∗	∗	NOUN
ejpam-4854	97	32	〈	〈	PROPN
ejpam-4854	97	33	erβz	erβz	NOUN
ejpam-4854	97	34	,	,	PUNCT
ejpam-4854	97	35	eβz	eβz	NOUN
ejpam-4854	97	36	−	−	PROPN
ejpam-4854	97	37	1	1	NUM
ejpam-4854	97	38	β	β	X
ejpam-4854	97	39	〉	〉	NOUN
ejpam-4854	97	40	=	=	SYM
ejpam-4854	97	41	〈	〈	PROPN
ejpam-4854	97	42	e(r−rβ)z+rβz	e(r−rβ)z+rβz	NOUN
ejpam-4854	97	43	,	,	PUNCT
ejpam-4854	97	44	eβz	eβz	PROPN
ejpam-4854	97	45	−	−	PROPN
ejpam-4854	97	46	1	1	NUM
ejpam-4854	97	47	β	β	X
ejpam-4854	97	48	〉	〉	NOUN
ejpam-4854	97	49	=	=	SYM
ejpam-4854	97	50	〈	〈	PROPN
ejpam-4854	97	51	erz−rβz+rβz	erz−rβz+rβz	PROPN
ejpam-4854	97	52	,	,	PUNCT
ejpam-4854	97	53	eβz	eβz	X
ejpam-4854	97	54	−	−	PROPN
ejpam-4854	97	55	1	1	NUM
ejpam-4854	97	56	β	β	X
ejpam-4854	97	57	〉	〉	NOUN
ejpam-4854	97	58	=	=	SYM
ejpam-4854	97	59	〈	〈	PROPN
ejpam-4854	97	60	erz	erz	PROPN
ejpam-4854	97	61	,	,	PUNCT
ejpam-4854	97	62	eβz	eβz	NOUN
ejpam-4854	97	63	−	−	PROPN
ejpam-4854	97	64	1	1	NUM
ejpam-4854	97	65	β	β	X
ejpam-4854	97	66	〉	〉	NOUN
ejpam-4854	97	67	=	=	SYM
ejpam-4854	97	68	s(β	s(β	PROPN
ejpam-4854	97	69	,	,	PUNCT
ejpam-4854	97	70	r)(n	r)(n	PROPN
ejpam-4854	97	71	)	)	PUNCT
ejpam-4854	97	72	example	example	NOUN
ejpam-4854	98	1	5	5	NUM
ejpam-4854	98	2	.	.	PUNCT
ejpam-4854	98	3	let	let	VERB
ejpam-4854	98	4	n	n	NOUN
ejpam-4854	98	5	=	=	SYM
ejpam-4854	98	6	4	4	X
ejpam-4854	98	7	.	.	PUNCT
ejpam-4854	98	8	then	then	ADV
ejpam-4854	98	9	p4[r	p4[r	VERB
ejpam-4854	98	10	−	−	PROPN
ejpam-4854	98	11	rβ]s(1,r)(4)[β	rβ]s(1,r)(4)[β	NOUN
ejpam-4854	98	12	]	]	X
ejpam-4854	98	13	=	=	SYM
ejpam-4854	98	14			NOUN
ejpam-4854	98	15	1	1	NUM
ejpam-4854	98	16	0	0	NUM
ejpam-4854	98	17	0	0	NUM
ejpam-4854	98	18	0	0	NUM
ejpam-4854	99	1	(	(	PUNCT
ejpam-4854	99	2	r	r	NOUN
ejpam-4854	99	3	−	−	NOUN
ejpam-4854	99	4	rβ	rβ	NOUN
ejpam-4854	99	5	)	)	PUNCT
ejpam-4854	99	6	1	1	NUM
ejpam-4854	99	7	0	0	NUM
ejpam-4854	99	8	0	0	NUM
ejpam-4854	100	1	(	(	PUNCT
ejpam-4854	100	2	r	r	NOUN
ejpam-4854	100	3	−	−	PROPN
ejpam-4854	100	4	rβ)2	rβ)2	NOUN
ejpam-4854	100	5	2(r	2(r	NUM
ejpam-4854	100	6	−	−	NOUN
ejpam-4854	100	7	rβ	rβ	NOUN
ejpam-4854	100	8	)	)	PUNCT
ejpam-4854	100	9	1	1	NUM
ejpam-4854	100	10	0	0	NUM
ejpam-4854	101	1	(	(	PUNCT
ejpam-4854	101	2	r	r	NOUN
ejpam-4854	101	3	−	−	PROPN
ejpam-4854	101	4	rβ)3	rβ)3	PROPN
ejpam-4854	101	5	3(r	3(r	NUM
ejpam-4854	101	6	−	−	ADP
ejpam-4854	102	1	rβ)2	rβ)2	NOUN
ejpam-4854	102	2	3(r	3(r	NUM
ejpam-4854	102	3	−	−	NOUN
ejpam-4854	102	4	rβ	rβ	NOUN
ejpam-4854	102	5	)	)	PUNCT
ejpam-4854	102	6	1	1	NUM
ejpam-4854	102	7			NOUN
ejpam-4854	102	8	·	·	PUNCT
ejpam-4854	102	9			NOUN
ejpam-4854	103	1	1	1	NUM
ejpam-4854	103	2	0	0	NUM
ejpam-4854	103	3	0	0	NUM
ejpam-4854	103	4	0	0	NUM
ejpam-4854	104	1	βr	βr	NUM
ejpam-4854	104	2	1	1	NUM
ejpam-4854	104	3	0	0	NUM
ejpam-4854	104	4	0	0	NUM
ejpam-4854	105	1	(	(	PUNCT
ejpam-4854	105	2	βr)2	βr)2	NOUN
ejpam-4854	105	3	β	β	X
ejpam-4854	106	1	+	+	CCONJ
ejpam-4854	106	2	2βr	2βr	ADJ
ejpam-4854	106	3	1	1	NUM
ejpam-4854	106	4	0	0	NUM
ejpam-4854	106	5	(	(	PUNCT
ejpam-4854	106	6	βr)3	βr)3	PROPN
ejpam-4854	106	7	β2	β2	NOUN
ejpam-4854	106	8	+	+	CCONJ
ejpam-4854	106	9	3β2r	3β2r	NUM
ejpam-4854	106	10	+	+	SYM
ejpam-4854	106	11	3(βr)2	3(βr)2	NUM
ejpam-4854	106	12	3β	3β	NOUN
ejpam-4854	107	1	+	+	CCONJ
ejpam-4854	107	2	3βr	3βr	ADJ
ejpam-4854	107	3	1	1	NUM
ejpam-4854	107	4			NOUN
ejpam-4854	107	5	=	=	PUNCT
ejpam-4854	107	6			NOUN
ejpam-4854	108	1	1	1	NUM
ejpam-4854	108	2	0	0	NUM
ejpam-4854	108	3	0	0	NUM
ejpam-4854	108	4	0	0	NUM
ejpam-4854	109	1	r	r	NOUN
ejpam-4854	109	2	1	1	NUM
ejpam-4854	109	3	0	0	NUM
ejpam-4854	109	4	0	0	NUM
ejpam-4854	109	5	r2	r2	NOUN
ejpam-4854	109	6	β	β	X
ejpam-4854	109	7	+	+	CCONJ
ejpam-4854	109	8	2r	2r	NUM
ejpam-4854	109	9	1	1	NUM
ejpam-4854	109	10	0	0	NUM
ejpam-4854	109	11	r3	r3	PROPN
ejpam-4854	109	12	β2	β2	NOUN
ejpam-4854	109	13	+	+	CCONJ
ejpam-4854	109	14	3βr	3βr	ADJ
ejpam-4854	109	15	+	+	CCONJ
ejpam-4854	109	16	3r2	3r2	NUM
ejpam-4854	109	17	3β	3β	NUM
ejpam-4854	109	18	+	+	CCONJ
ejpam-4854	109	19	3r	3r	NUM
ejpam-4854	109	20	1	1	NUM
ejpam-4854	109	21			NOUN
ejpam-4854	109	22	=	=	SYM
ejpam-4854	109	23	s(β	s(β	PROPN
ejpam-4854	109	24	,	,	PUNCT
ejpam-4854	109	25	r)(4	r)(4	NUM
ejpam-4854	109	26	)	)	PUNCT
ejpam-4854	109	27	.	.	PUNCT
ejpam-4854	110	1	2.2	2.2	NUM
ejpam-4854	110	2	.	.	PUNCT
ejpam-4854	110	3	factorization	factorization	NOUN
ejpam-4854	110	4	of	of	ADP
ejpam-4854	110	5	the	the	DET
ejpam-4854	110	6	(	(	PUNCT
ejpam-4854	110	7	r	r	NOUN
ejpam-4854	110	8	,	,	PUNCT
ejpam-4854	110	9	β)-stirling	β)-stirle	VERB
ejpam-4854	110	10	matrix	matrix	NOUN
ejpam-4854	110	11	to	to	PART
ejpam-4854	110	12	factor	factor	VERB
ejpam-4854	110	13	the	the	DET
ejpam-4854	110	14	(	(	PUNCT
ejpam-4854	110	15	r	r	NOUN
ejpam-4854	110	16	,	,	PUNCT
ejpam-4854	110	17	β)-stirling	β)-stirle	VERB
ejpam-4854	110	18	matrix	matrix	NOUN
ejpam-4854	110	19	,	,	PUNCT
ejpam-4854	110	20	we	we	PRON
ejpam-4854	110	21	need	need	VERB
ejpam-4854	110	22	the	the	DET
ejpam-4854	110	23	following	follow	VERB
ejpam-4854	110	24	matrices	matrix	NOUN
ejpam-4854	110	25	defined	define	VERB
ejpam-4854	110	26	by	by	ADP
ejpam-4854	110	27	zhang	zhang	PROPN
ejpam-4854	110	28	in	in	ADP
ejpam-4854	110	29	[	[	X
ejpam-4854	110	30	13	13	NUM
ejpam-4854	110	31	]	]	PUNCT
ejpam-4854	110	32	:	:	PUNCT
ejpam-4854	110	33	sn[x	sn[x	PUNCT
ejpam-4854	110	34	]	]	X
ejpam-4854	111	1	=	=	X
ejpam-4854	111	2	[	[	PUNCT
ejpam-4854	111	3	xi−j	xi−j	PROPN
ejpam-4854	111	4	]	]	PUNCT
ejpam-4854	111	5	0≤j	0≤j	PROPN
ejpam-4854	111	6	,	,	PUNCT
ejpam-4854	111	7	i≤n−1	i≤n−1	PROPN
ejpam-4854	111	8	for	for	ADP
ejpam-4854	111	9	example	example	NOUN
ejpam-4854	111	10	,	,	PUNCT
ejpam-4854	111	11	when	when	SCONJ
ejpam-4854	111	12	n	n	X
ejpam-4854	111	13	=	=	SYM
ejpam-4854	111	14	4	4	NUM
ejpam-4854	111	15	,	,	PUNCT
ejpam-4854	111	16	s4[x	s4[x	NOUN
ejpam-4854	111	17	]	]	PUNCT
ejpam-4854	111	18	=	=	SYM
ejpam-4854	111	19			NOUN
ejpam-4854	111	20	1	1	NUM
ejpam-4854	111	21	0	0	NUM
ejpam-4854	111	22	0	0	NUM
ejpam-4854	111	23	0	0	NUM
ejpam-4854	111	24	x	x	SYM
ejpam-4854	111	25	1	1	NUM
ejpam-4854	111	26	0	0	NUM
ejpam-4854	111	27	0	0	NUM
ejpam-4854	112	1	x2	x2	NOUN
ejpam-4854	112	2	x	x	SYM
ejpam-4854	112	3	1	1	NUM
ejpam-4854	112	4	0	0	NUM
ejpam-4854	113	1	x3	x3	ADJ
ejpam-4854	113	2	x2	x2	NOUN
ejpam-4854	113	3	x	x	SYM
ejpam-4854	113	4	1	1	NUM
ejpam-4854	113	5			NOUN
ejpam-4854	113	6	.	.	PUNCT
ejpam-4854	114	1	zhang	zhang	PROPN
ejpam-4854	114	2	also	also	ADV
ejpam-4854	114	3	define	define	VERB
ejpam-4854	114	4	the	the	DET
ejpam-4854	114	5	n×	n×	PROPN
ejpam-4854	114	6	n	n	NOUN
ejpam-4854	114	7	matrix	matrix	NOUN
ejpam-4854	114	8	gk[x	gk[x	NOUN
ejpam-4854	114	9	]	]	PUNCT
ejpam-4854	114	10	as	as	ADP
ejpam-4854	114	11	gk[x	gk[x	PROPN
ejpam-4854	114	12	]	]	X
ejpam-4854	114	13	=	=	SYM
ejpam-4854	114	14	in−k	in−k	PROPN
ejpam-4854	114	15	⊕	⊕	PROPN
ejpam-4854	114	16	sk[x	sk[x	PROPN
ejpam-4854	114	17	]	]	PUNCT
ejpam-4854	114	18	,	,	PUNCT
ejpam-4854	114	19	(	(	PUNCT
ejpam-4854	114	20	1	1	NUM
ejpam-4854	114	21	≤	≤	NUM
ejpam-4854	114	22	k	k	X
ejpam-4854	114	23	≤	≤	PROPN
ejpam-4854	114	24	n−	n−	NOUN
ejpam-4854	114	25	1	1	NUM
ejpam-4854	114	26	)	)	PUNCT
ejpam-4854	114	27	,	,	PUNCT
ejpam-4854	114	28	g.	g.	PROPN
ejpam-4854	114	29	engalan	engalan	PROPN
ejpam-4854	114	30	,	,	PUNCT
ejpam-4854	114	31	m.r	m.r	PROPN
ejpam-4854	114	32	.	.	PROPN
ejpam-4854	114	33	latayada	latayada	PROPN
ejpam-4854	114	34	/	/	SYM
ejpam-4854	114	35	eur	eur	PROPN
ejpam-4854	114	36	.	.	PUNCT
ejpam-4854	115	1	j.	j.	PROPN
ejpam-4854	115	2	pure	pure	PROPN
ejpam-4854	115	3	appl	appl	PROPN
ejpam-4854	115	4	.	.	PROPN
ejpam-4854	115	5	math	math	PROPN
ejpam-4854	115	6	,	,	PUNCT
ejpam-4854	115	7	16	16	NUM
ejpam-4854	115	8	(	(	PUNCT
ejpam-4854	115	9	4	4	NUM
ejpam-4854	115	10	)	)	PUNCT
ejpam-4854	115	11	(	(	PUNCT
ejpam-4854	115	12	2023	2023	NUM
ejpam-4854	115	13	)	)	PUNCT
ejpam-4854	115	14	,	,	PUNCT
ejpam-4854	115	15	2306	2306	NUM
ejpam-4854	115	16	-	-	SYM
ejpam-4854	115	17	2322	2322	NUM
ejpam-4854	115	18	2312	2312	NUM
ejpam-4854	115	19	where	where	SCONJ
ejpam-4854	115	20	gn[k	gn[k	PROPN
ejpam-4854	115	21	]	]	X
ejpam-4854	115	22	=	=	SYM
ejpam-4854	115	23	sn[k	sn[k	PROPN
ejpam-4854	115	24	]	]	PUNCT
ejpam-4854	115	25	and	and	CCONJ
ejpam-4854	115	26	⊕	⊕	PROPN
ejpam-4854	115	27	denotes	denote	VERB
ejpam-4854	115	28	the	the	DET
ejpam-4854	115	29	matrix	matrix	NOUN
ejpam-4854	115	30	direct	direct	ADJ
ejpam-4854	115	31	sum	sum	NOUN
ejpam-4854	115	32	.	.	PUNCT
ejpam-4854	116	1	proposition	proposition	NOUN
ejpam-4854	116	2	5	5	NUM
ejpam-4854	116	3	.	.	PUNCT
ejpam-4854	117	1	for	for	ADP
ejpam-4854	117	2	any	any	DET
ejpam-4854	117	3	integer	integer	NOUN
ejpam-4854	117	4	n	n	CCONJ
ejpam-4854	117	5	,	,	PUNCT
ejpam-4854	117	6	m	m	VERB
ejpam-4854	117	7	≥	≥	NOUN
ejpam-4854	117	8	1	1	NUM
ejpam-4854	117	9	and	and	CCONJ
ejpam-4854	117	10	r	r	NOUN
ejpam-4854	117	11	≥	≥	NOUN
ejpam-4854	117	12	0	0	NUM
ejpam-4854	117	13	,	,	PUNCT
ejpam-4854	117	14	we	we	PRON
ejpam-4854	117	15	have	have	VERB
ejpam-4854	117	16	s(β	s(β	PROPN
ejpam-4854	117	17	,	,	PUNCT
ejpam-4854	117	18	r)(n	r)(n	PROPN
ejpam-4854	117	19	)	)	PUNCT
ejpam-4854	118	1	=	=	SYM
ejpam-4854	118	2	gn[r]gn−1[r	gn[r]gn−1[r	X
ejpam-4854	118	3	]	]	X
ejpam-4854	118	4	·	·	PUNCT
ejpam-4854	118	5	·	·	PUNCT
ejpam-4854	118	6	·	·	PUNCT
ejpam-4854	118	7	g1[r]p̄n−1[β	g1[r]p̄n−1[β	X
ejpam-4854	118	8	]	]	X
ejpam-4854	118	9	·	·	PUNCT
ejpam-4854	118	10	·	·	PUNCT
ejpam-4854	118	11	·	·	PUNCT
ejpam-4854	119	1	p̄1[β	p̄1[β	NUM
ejpam-4854	119	2	]	]	X
ejpam-4854	119	3	proof	proof	NOUN
ejpam-4854	119	4	.	.	PUNCT
ejpam-4854	120	1	by	by	ADP
ejpam-4854	120	2	proposition	proposition	NOUN
ejpam-4854	120	3	3	3	NUM
ejpam-4854	120	4	,	,	PUNCT
ejpam-4854	120	5	s(β	s(β	PROPN
ejpam-4854	120	6	,	,	PUNCT
ejpam-4854	120	7	r	r	NOUN
ejpam-4854	120	8	)	)	PUNCT
ejpam-4854	120	9	=	=	SYM
ejpam-4854	120	10	pn[r]s2(n)[β	pn[r]s2(n)[β	PROPN
ejpam-4854	120	11	]	]	PUNCT
ejpam-4854	120	12	.	.	PUNCT
ejpam-4854	121	1	note	note	VERB
ejpam-4854	121	2	that	that	SCONJ
ejpam-4854	121	3	by	by	ADP
ejpam-4854	121	4	theorem	theorem	NOUN
ejpam-4854	121	5	1	1	NUM
ejpam-4854	121	6	of	of	ADP
ejpam-4854	121	7	[	[	X
ejpam-4854	121	8	13	13	NUM
ejpam-4854	121	9	]	]	PUNCT
ejpam-4854	121	10	,	,	PUNCT
ejpam-4854	121	11	pn[r	pn[r	NOUN
ejpam-4854	121	12	]	]	X
ejpam-4854	121	13	=	=	SYM
ejpam-4854	121	14	gn[r]gn−1[r	gn[r]gn−1[r	X
ejpam-4854	121	15	]	]	X
ejpam-4854	121	16	·	·	PUNCT
ejpam-4854	121	17	·	·	PUNCT
ejpam-4854	121	18	·	·	PUNCT
ejpam-4854	121	19	g1[r	g1[r	NOUN
ejpam-4854	121	20	]	]	PUNCT
ejpam-4854	121	21	.	.	PUNCT
ejpam-4854	122	1	also	also	ADV
ejpam-4854	122	2	,	,	PUNCT
ejpam-4854	122	3	based	base	VERB
ejpam-4854	122	4	on	on	ADP
ejpam-4854	122	5	one	one	NUM
ejpam-4854	122	6	of	of	ADP
ejpam-4854	122	7	the	the	DET
ejpam-4854	122	8	results	result	NOUN
ejpam-4854	122	9	of	of	ADP
ejpam-4854	122	10	cheon	cheon	PROPN
ejpam-4854	122	11	and	and	CCONJ
ejpam-4854	122	12	kim	kim	PROPN
ejpam-4854	122	13	in	in	ADP
ejpam-4854	122	14	[	[	X
ejpam-4854	122	15	3	3	NUM
ejpam-4854	122	16	]	]	PUNCT
ejpam-4854	122	17	,	,	PUNCT
ejpam-4854	122	18	s2(n)[β	s2(n)[β	PROPN
ejpam-4854	122	19	]	]	X
ejpam-4854	122	20	=	=	SYM
ejpam-4854	122	21	p̄n−1[β	p̄n−1[β	NOUN
ejpam-4854	122	22	]	]	PUNCT
ejpam-4854	122	23	·	·	PUNCT
ejpam-4854	122	24	·	·	PUNCT
ejpam-4854	122	25	·	·	PUNCT
ejpam-4854	122	26	p̄1[β	p̄1[β	PROPN
ejpam-4854	122	27	]	]	PUNCT
ejpam-4854	122	28	.	.	PUNCT
ejpam-4854	123	1	this	this	PRON
ejpam-4854	123	2	follows	follow	VERB
ejpam-4854	123	3	that	that	SCONJ
ejpam-4854	123	4	,	,	PUNCT
ejpam-4854	123	5	s(β	s(β	PROPN
ejpam-4854	123	6	,	,	PUNCT
ejpam-4854	123	7	r)(n	r)(n	PROPN
ejpam-4854	123	8	)	)	PUNCT
ejpam-4854	123	9	=	=	SYM
ejpam-4854	123	10	gn[r]gn−1[r	gn[r]gn−1[r	X
ejpam-4854	123	11	]	]	X
ejpam-4854	123	12	·	·	PUNCT
ejpam-4854	123	13	·	·	PUNCT
ejpam-4854	123	14	·	·	PUNCT
ejpam-4854	123	15	g1[r]p̄n−1[β	g1[r]p̄n−1[β	X
ejpam-4854	123	16	]	]	X
ejpam-4854	123	17	·	·	PUNCT
ejpam-4854	123	18	·	·	PUNCT
ejpam-4854	123	19	·	·	PUNCT
ejpam-4854	124	1	p̄1[β	p̄1[β	PROPN
ejpam-4854	124	2	]	]	PUNCT
ejpam-4854	124	3	.	.	PUNCT
ejpam-4854	125	1	2.3	2.3	NUM
ejpam-4854	125	2	.	.	PUNCT
ejpam-4854	125	3	relationship	relationship	NOUN
ejpam-4854	125	4	between	between	ADP
ejpam-4854	125	5	the	the	DET
ejpam-4854	125	6	(	(	PUNCT
ejpam-4854	125	7	r	r	NOUN
ejpam-4854	125	8	,	,	PUNCT
ejpam-4854	125	9	β)-stirling	β)-stirle	VERB
ejpam-4854	125	10	matrix	matrix	NOUN
ejpam-4854	125	11	and	and	CCONJ
ejpam-4854	125	12	a	a	DET
ejpam-4854	125	13	generalized	generalized	ADJ
ejpam-4854	125	14	vandermonde	vandermonde	NOUN
ejpam-4854	125	15	matrix	matrix	NOUN
ejpam-4854	125	16	in	in	ADP
ejpam-4854	125	17	this	this	DET
ejpam-4854	125	18	section	section	NOUN
ejpam-4854	125	19	,	,	PUNCT
ejpam-4854	125	20	we	we	PRON
ejpam-4854	125	21	introduce	introduce	VERB
ejpam-4854	125	22	a	a	DET
ejpam-4854	125	23	generalization	generalization	NOUN
ejpam-4854	125	24	of	of	ADP
ejpam-4854	125	25	the	the	DET
ejpam-4854	125	26	vandermonde	vandermonde	ADJ
ejpam-4854	125	27	matrix	matrix	NOUN
ejpam-4854	125	28	which	which	PRON
ejpam-4854	125	29	will	will	AUX
ejpam-4854	125	30	be	be	AUX
ejpam-4854	125	31	useful	useful	ADJ
ejpam-4854	125	32	in	in	ADP
ejpam-4854	125	33	the	the	DET
ejpam-4854	125	34	study	study	NOUN
ejpam-4854	125	35	of	of	ADP
ejpam-4854	125	36	successive	successive	ADJ
ejpam-4854	125	37	power	power	NOUN
ejpam-4854	125	38	sums	sum	NOUN
ejpam-4854	125	39	of	of	ADP
ejpam-4854	125	40	arithmetic	arithmetic	ADJ
ejpam-4854	125	41	progression	progression	NOUN
ejpam-4854	125	42	.	.	PUNCT
ejpam-4854	126	1	to	to	PART
ejpam-4854	126	2	do	do	VERB
ejpam-4854	126	3	that	that	PRON
ejpam-4854	126	4	,	,	PUNCT
ejpam-4854	126	5	we	we	PRON
ejpam-4854	126	6	define	define	VERB
ejpam-4854	126	7	the	the	DET
ejpam-4854	126	8	following	follow	VERB
ejpam-4854	126	9	matrices	matrix	NOUN
ejpam-4854	126	10	.	.	PUNCT
ejpam-4854	127	1	definition	definition	NOUN
ejpam-4854	127	2	2	2	NUM
ejpam-4854	127	3	.	.	PUNCT
ejpam-4854	128	1	let	let	VERB
ejpam-4854	128	2	s(β	s(β	PROPN
ejpam-4854	128	3	,	,	PUNCT
ejpam-4854	128	4	r)(n	r)(n	PROPN
ejpam-4854	128	5	)	)	PUNCT
ejpam-4854	128	6	be	be	VERB
ejpam-4854	128	7	the	the	DET
ejpam-4854	128	8	(	(	PUNCT
ejpam-4854	128	9	r	r	NOUN
ejpam-4854	128	10	,	,	PUNCT
ejpam-4854	128	11	β)-stirling	β)-stirle	VERB
ejpam-4854	128	12	matrix	matrix	NOUN
ejpam-4854	128	13	.	.	PUNCT
ejpam-4854	129	1	the	the	DET
ejpam-4854	129	2	matrix	matrix	NOUN
ejpam-4854	129	3	factorial	factorial	NOUN
ejpam-4854	129	4	of	of	ADP
ejpam-4854	129	5	the	the	DET
ejpam-4854	129	6	(	(	PUNCT
ejpam-4854	129	7	r	r	NOUN
ejpam-4854	129	8	,	,	PUNCT
ejpam-4854	129	9	β)-stirling	β)-stirle	VERB
ejpam-4854	129	10	matrix	matrix	NOUN
ejpam-4854	129	11	,	,	PUNCT
ejpam-4854	129	12	denoted	denote	VERB
ejpam-4854	129	13	by	by	ADP
ejpam-4854	129	14	s̃(β	s̃(β	PROPN
ejpam-4854	129	15	,	,	PUNCT
ejpam-4854	129	16	r)(n	r)(n	PROPN
ejpam-4854	129	17	)	)	PUNCT
ejpam-4854	129	18	is	be	AUX
ejpam-4854	129	19	defined	define	VERB
ejpam-4854	129	20	by	by	ADP
ejpam-4854	129	21	s̃(β	s̃(β	PROPN
ejpam-4854	129	22	,	,	PUNCT
ejpam-4854	129	23	r)(n	r)(n	PROPN
ejpam-4854	129	24	)	)	PUNCT
ejpam-4854	129	25	:	:	PUNCT
ejpam-4854	130	1	=	=	PUNCT
ejpam-4854	130	2	s(β	s(β	PROPN
ejpam-4854	130	3	,	,	PUNCT
ejpam-4854	130	4	r)(n	r)(n	PROPN
ejpam-4854	130	5	)	)	PUNCT
ejpam-4854	130	6	·	·	PUNCT
ejpam-4854	130	7	diag(0	diag(0	NOUN
ejpam-4854	130	8	!	!	PUNCT
ejpam-4854	130	9	,	,	PUNCT
ejpam-4854	130	10	1	1	X
ejpam-4854	130	11	!	!	NUM
ejpam-4854	130	12	,	,	PUNCT
ejpam-4854	130	13	.	.	PUNCT
ejpam-4854	130	14	.	.	PUNCT
ejpam-4854	130	15	.	.	PUNCT
ejpam-4854	130	16	,	,	PUNCT
ejpam-4854	130	17	n	n	CCONJ
ejpam-4854	130	18	!	!	PUNCT
ejpam-4854	130	19	)	)	PUNCT
ejpam-4854	130	20	.	.	PUNCT
ejpam-4854	131	1	(	(	PUNCT
ejpam-4854	131	2	4	4	X
ejpam-4854	131	3	)	)	PUNCT
ejpam-4854	131	4	example	example	NOUN
ejpam-4854	131	5	6	6	NUM
ejpam-4854	131	6	.	.	PUNCT
ejpam-4854	132	1	let	let	VERB
ejpam-4854	132	2	n	n	NOUN
ejpam-4854	132	3	=	=	SYM
ejpam-4854	132	4	4	4	X
ejpam-4854	132	5	.	.	PUNCT
ejpam-4854	133	1	then	then	ADV
ejpam-4854	133	2	s̃(β	s̃(β	PROPN
ejpam-4854	133	3	,	,	PUNCT
ejpam-4854	133	4	r)(4	r)(4	NUM
ejpam-4854	133	5	)	)	PUNCT
ejpam-4854	133	6	=	=	SYM
ejpam-4854	133	7	s(β	s(β	PROPN
ejpam-4854	133	8	,	,	PUNCT
ejpam-4854	133	9	r)(4	r)(4	NUM
ejpam-4854	133	10	)	)	PUNCT
ejpam-4854	133	11	·	·	PUNCT
ejpam-4854	134	1	diag(0	diag(0	NOUN
ejpam-4854	134	2	!	!	PUNCT
ejpam-4854	134	3	,	,	PUNCT
ejpam-4854	134	4	1	1	X
ejpam-4854	134	5	!	!	NUM
ejpam-4854	134	6	,	,	PUNCT
ejpam-4854	134	7	2	2	X
ejpam-4854	134	8	!	!	NUM
ejpam-4854	134	9	,	,	PUNCT
ejpam-4854	134	10	3	3	NUM
ejpam-4854	134	11	!	!	PUNCT
ejpam-4854	134	12	)	)	PUNCT
ejpam-4854	135	1	=	=	SYM
ejpam-4854	135	2			NOUN
ejpam-4854	135	3	0	0	NUM
ejpam-4854	135	4	!	!	SYM
ejpam-4854	135	5	0	0	NUM
ejpam-4854	135	6	0	0	NUM
ejpam-4854	135	7	0	0	NUM
ejpam-4854	136	1	r	r	NOUN
ejpam-4854	136	2	1	1	NUM
ejpam-4854	136	3	!	!	NOUN
ejpam-4854	136	4	0	0	NUM
ejpam-4854	136	5	0	0	NUM
ejpam-4854	136	6	r2	r2	PROPN
ejpam-4854	136	7	β	β	X
ejpam-4854	136	8	+	+	X
ejpam-4854	136	9	2r	2r	NUM
ejpam-4854	136	10	2	2	NUM
ejpam-4854	136	11	!	!	SYM
ejpam-4854	136	12	0	0	NUM
ejpam-4854	136	13	r3	r3	PROPN
ejpam-4854	136	14	β2	β2	NOUN
ejpam-4854	136	15	+	+	CCONJ
ejpam-4854	136	16	3βr	3βr	ADJ
ejpam-4854	136	17	+	+	CCONJ
ejpam-4854	136	18	3r2	3r2	NUM
ejpam-4854	136	19	(	(	PUNCT
ejpam-4854	136	20	3β	3β	NUM
ejpam-4854	136	21	+	+	CCONJ
ejpam-4854	136	22	3r)2	3r)2	NUM
ejpam-4854	136	23	!	!	PUNCT
ejpam-4854	137	1	3	3	X
ejpam-4854	137	2	!	!	X
ejpam-4854	137	3			NOUN
ejpam-4854	137	4	=	=	SYM
ejpam-4854	137	5			NOUN
ejpam-4854	137	6	1	1	NUM
ejpam-4854	137	7	0	0	NUM
ejpam-4854	137	8	0	0	NUM
ejpam-4854	137	9	0	0	NUM
ejpam-4854	137	10	r	r	NOUN
ejpam-4854	137	11	1	1	NUM
ejpam-4854	137	12	0	0	NUM
ejpam-4854	137	13	0	0	NUM
ejpam-4854	137	14	r2	r2	NOUN
ejpam-4854	137	15	β	β	X
ejpam-4854	137	16	+	+	CCONJ
ejpam-4854	137	17	2r	2r	NUM
ejpam-4854	137	18	2	2	NUM
ejpam-4854	137	19	0	0	NUM
ejpam-4854	137	20	r3	r3	PROPN
ejpam-4854	137	21	β2	β2	NOUN
ejpam-4854	137	22	+	+	CCONJ
ejpam-4854	137	23	3βr	3βr	ADJ
ejpam-4854	137	24	+	+	CCONJ
ejpam-4854	137	25	3r2	3r2	NUM
ejpam-4854	137	26	6β	6β	NOUN
ejpam-4854	137	27	+	+	CCONJ
ejpam-4854	137	28	6r	6r	NUM
ejpam-4854	137	29	6	6	NUM
ejpam-4854	137	30			NOUN
ejpam-4854	137	31	.	.	PUNCT
ejpam-4854	138	1	g.	g.	PROPN
ejpam-4854	138	2	engalan	engalan	PROPN
ejpam-4854	138	3	,	,	PUNCT
ejpam-4854	138	4	m.r	m.r	PROPN
ejpam-4854	138	5	.	.	PROPN
ejpam-4854	138	6	latayada	latayada	PROPN
ejpam-4854	138	7	/	/	SYM
ejpam-4854	138	8	eur	eur	PROPN
ejpam-4854	138	9	.	.	PUNCT
ejpam-4854	139	1	j.	j.	PROPN
ejpam-4854	139	2	pure	pure	PROPN
ejpam-4854	139	3	appl	appl	PROPN
ejpam-4854	139	4	.	.	PROPN
ejpam-4854	139	5	math	math	PROPN
ejpam-4854	139	6	,	,	PUNCT
ejpam-4854	139	7	16	16	NUM
ejpam-4854	139	8	(	(	PUNCT
ejpam-4854	139	9	4	4	NUM
ejpam-4854	139	10	)	)	PUNCT
ejpam-4854	139	11	(	(	PUNCT
ejpam-4854	139	12	2023	2023	NUM
ejpam-4854	139	13	)	)	PUNCT
ejpam-4854	139	14	,	,	PUNCT
ejpam-4854	139	15	2306	2306	NUM
ejpam-4854	139	16	-	-	SYM
ejpam-4854	139	17	2322	2322	NUM
ejpam-4854	139	18	2313	2313	NUM
ejpam-4854	139	19	theorem	theorem	NOUN
ejpam-4854	139	20	1	1	NUM
ejpam-4854	139	21	.	.	PUNCT
ejpam-4854	140	1	let	let	VERB
ejpam-4854	140	2	vβ	vβ	NOUN
ejpam-4854	140	3	,	,	PUNCT
ejpam-4854	140	4	r	r	NOUN
ejpam-4854	140	5	n	n	PROPN
ejpam-4854	140	6	(	(	PUNCT
ejpam-4854	140	7	t	t	NOUN
ejpam-4854	140	8	)	)	PUNCT
ejpam-4854	140	9	be	be	VERB
ejpam-4854	140	10	the	the	DET
ejpam-4854	140	11	n×	n×	PROPN
ejpam-4854	140	12	n	n	CCONJ
ejpam-4854	140	13	generalized	generalized	ADJ
ejpam-4854	140	14	vandermonde	vandermonde	NOUN
ejpam-4854	140	15	matrix	matrix	NOUN
ejpam-4854	140	16	defined	define	VERB
ejpam-4854	140	17	by	by	ADP
ejpam-4854	140	18	vβ	vβ	X
ejpam-4854	140	19	,	,	PUNCT
ejpam-4854	140	20	r	r	NOUN
ejpam-4854	140	21	n	n	PROPN
ejpam-4854	140	22	(	(	PUNCT
ejpam-4854	140	23	t	t	PROPN
ejpam-4854	140	24	)	)	PUNCT
ejpam-4854	140	25	:	:	PUNCT
ejpam-4854	141	1	=	=	SYM
ejpam-4854	141	2	vβ	vβ	X
ejpam-4854	141	3	,	,	PUNCT
ejpam-4854	141	4	r	r	NOUN
ejpam-4854	141	5	n	n	NUM
ejpam-4854	141	6	(	(	PUNCT
ejpam-4854	141	7	βt+	βt+	ADJ
ejpam-4854	141	8	r	r	NOUN
ejpam-4854	141	9	,	,	PUNCT
ejpam-4854	141	10	βt+	βt+	ADJ
ejpam-4854	141	11	β	β	X
ejpam-4854	141	12	+	+	CCONJ
ejpam-4854	141	13	r	r	NOUN
ejpam-4854	141	14	,	,	PUNCT
ejpam-4854	141	15	βt+	βt+	ADJ
ejpam-4854	141	16	2β	2β	NOUN
ejpam-4854	141	17	+	+	CCONJ
ejpam-4854	141	18	r	r	NOUN
ejpam-4854	141	19	,	,	PUNCT
ejpam-4854	141	20	.	.	PUNCT
ejpam-4854	141	21	.	.	PUNCT
ejpam-4854	141	22	.	.	PUNCT
ejpam-4854	142	1	,	,	PUNCT
ejpam-4854	142	2	βt+	βt+	X
ejpam-4854	142	3	(	(	PUNCT
ejpam-4854	142	4	n−	n−	NOUN
ejpam-4854	142	5	1)β	1)β	NUM
ejpam-4854	142	6	+	+	CCONJ
ejpam-4854	142	7	r	r	NOUN
ejpam-4854	142	8	)	)	PUNCT
ejpam-4854	142	9	=	=	PRON
ejpam-4854	142	10			VERB
ejpam-4854	142	11	1	1	NUM
ejpam-4854	142	12	1	1	NUM
ejpam-4854	142	13	1	1	NUM
ejpam-4854	142	14	·	·	PUNCT
ejpam-4854	142	15	·	·	PUNCT
ejpam-4854	142	16	·	·	PUNCT
ejpam-4854	143	1	1	1	NUM
ejpam-4854	143	2	βt+	βt+	ADJ
ejpam-4854	143	3	r	r	NOUN
ejpam-4854	143	4	βt+	βt+	X
ejpam-4854	143	5	β	β	X
ejpam-4854	144	1	+	+	NOUN
ejpam-4854	144	2	r	r	NOUN
ejpam-4854	144	3	βt+	βt+	ADJ
ejpam-4854	144	4	2β	2β	NOUN
ejpam-4854	144	5	+	+	CCONJ
ejpam-4854	144	6	r	r	NOUN
ejpam-4854	144	7	·	·	PUNCT
ejpam-4854	144	8	·	·	PUNCT
ejpam-4854	144	9	·	·	PUNCT
ejpam-4854	144	10	βt+	βt+	ADJ
ejpam-4854	145	1	(	(	PUNCT
ejpam-4854	145	2	n−	n−	NOUN
ejpam-4854	145	3	1)β	1)β	NUM
ejpam-4854	145	4	+	+	CCONJ
ejpam-4854	145	5	r	r	NOUN
ejpam-4854	145	6	(	(	PUNCT
ejpam-4854	145	7	βt+	βt+	ADJ
ejpam-4854	145	8	r)2	r)2	NOUN
ejpam-4854	145	9	(	(	PUNCT
ejpam-4854	145	10	βt+	βt+	ADJ
ejpam-4854	145	11	β	β	X
ejpam-4854	146	1	+	+	ADJ
ejpam-4854	146	2	r)2	r)2	NOUN
ejpam-4854	146	3	(	(	PUNCT
ejpam-4854	146	4	βt+	βt+	ADJ
ejpam-4854	146	5	2β	2β	NOUN
ejpam-4854	146	6	+	+	CCONJ
ejpam-4854	146	7	r)2	r)2	ADJ
ejpam-4854	146	8	·	·	PUNCT
ejpam-4854	146	9	·	·	PUNCT
ejpam-4854	146	10	·	·	PUNCT
ejpam-4854	146	11	(	(	PUNCT
ejpam-4854	146	12	βt+	βt+	X
ejpam-4854	146	13	(	(	PUNCT
ejpam-4854	146	14	n−	n−	NOUN
ejpam-4854	146	15	1)β	1)β	NUM
ejpam-4854	146	16	+	+	CCONJ
ejpam-4854	146	17	r)2	r)2	NOUN
ejpam-4854	146	18	...	...	PUNCT
ejpam-4854	146	19	...	...	PUNCT
ejpam-4854	146	20	...	...	PUNCT
ejpam-4854	146	21	.	.	PUNCT
ejpam-4854	146	22	.	.	PUNCT
ejpam-4854	146	23	.	.	PUNCT
ejpam-4854	147	1	...	...	PUNCT
ejpam-4854	148	1	(	(	PUNCT
ejpam-4854	148	2	βt+	βt+	X
ejpam-4854	148	3	r)n−1	r)n−1	X
ejpam-4854	148	4	(	(	PUNCT
ejpam-4854	148	5	βt+	βt+	ADJ
ejpam-4854	148	6	β	β	X
ejpam-4854	148	7	+	+	X
ejpam-4854	148	8	r)n−1	r)n−1	ADJ
ejpam-4854	148	9	(	(	PUNCT
ejpam-4854	148	10	βt+	βt+	ADJ
ejpam-4854	148	11	2β	2β	NOUN
ejpam-4854	148	12	+	+	CCONJ
ejpam-4854	148	13	r)n−1	r)n−1	VERB
ejpam-4854	148	14	·	·	PUNCT
ejpam-4854	148	15	·	·	PUNCT
ejpam-4854	148	16	·	·	PUNCT
ejpam-4854	148	17	(	(	PUNCT
ejpam-4854	148	18	βt+	βt+	X
ejpam-4854	148	19	(	(	PUNCT
ejpam-4854	148	20	n−	n−	NOUN
ejpam-4854	148	21	1)β	1)β	NUM
ejpam-4854	148	22	+	+	CCONJ
ejpam-4854	148	23	r)n−1	r)n−1	ADJ
ejpam-4854	148	24			NOUN
ejpam-4854	148	25	and	and	CCONJ
ejpam-4854	148	26	cβ	cβ	NOUN
ejpam-4854	148	27	n	n	PROPN
ejpam-4854	148	28	(	(	PUNCT
ejpam-4854	148	29	t	t	PROPN
ejpam-4854	148	30	)	)	PUNCT
ejpam-4854	148	31	=	=	PUNCT
ejpam-4854	149	1	[	[	PUNCT
ejpam-4854	149	2	βi	βi	X
ejpam-4854	149	3	(	(	PUNCT
ejpam-4854	149	4	t+	t+	NOUN
ejpam-4854	149	5	j	j	PROPN
ejpam-4854	149	6	i	i	PROPN
ejpam-4854	149	7	)	)	PUNCT
ejpam-4854	149	8	]	]	PUNCT
ejpam-4854	149	9	0≤i	0≤i	PROPN
ejpam-4854	149	10	,	,	PUNCT
ejpam-4854	149	11	j≤n−1	j≤n−1	PROPN
ejpam-4854	149	12	.	.	PUNCT
ejpam-4854	150	1	then	then	ADV
ejpam-4854	150	2	we	we	PRON
ejpam-4854	150	3	can	can	AUX
ejpam-4854	150	4	factor	factor	VERB
ejpam-4854	150	5	vβ	vβ	NOUN
ejpam-4854	150	6	,	,	PUNCT
ejpam-4854	150	7	r	r	NOUN
ejpam-4854	150	8	n	n	PROPN
ejpam-4854	150	9	(	(	PUNCT
ejpam-4854	150	10	t	t	PROPN
ejpam-4854	150	11	)	)	PUNCT
ejpam-4854	150	12	as	as	ADP
ejpam-4854	150	13	vβ	vβ	X
ejpam-4854	150	14	,	,	PUNCT
ejpam-4854	150	15	r	r	NOUN
ejpam-4854	150	16	n	n	PROPN
ejpam-4854	150	17	(	(	PUNCT
ejpam-4854	150	18	t	t	PROPN
ejpam-4854	150	19	)	)	PUNCT
ejpam-4854	150	20	=	=	SYM
ejpam-4854	151	1	s̃(β	s̃(β	PROPN
ejpam-4854	151	2	,	,	PUNCT
ejpam-4854	151	3	r)(n)cβ	r)(n)cβ	NOUN
ejpam-4854	151	4	n	n	CCONJ
ejpam-4854	151	5	(	(	PUNCT
ejpam-4854	151	6	t	t	PROPN
ejpam-4854	151	7	)	)	PUNCT
ejpam-4854	151	8	.	.	PUNCT
ejpam-4854	152	1	(	(	PUNCT
ejpam-4854	152	2	5	5	X
ejpam-4854	152	3	)	)	PUNCT
ejpam-4854	152	4	proof	proof	NOUN
ejpam-4854	152	5	.	.	PUNCT
ejpam-4854	153	1	consider	consider	VERB
ejpam-4854	153	2	the	the	DET
ejpam-4854	153	3	equation	equation	NOUN
ejpam-4854	153	4	(	(	PUNCT
ejpam-4854	153	5	1	1	NUM
ejpam-4854	153	6	)	)	PUNCT
ejpam-4854	153	7	,	,	PUNCT
ejpam-4854	153	8	tn	tn	PROPN
ejpam-4854	153	9	=	=	SYM
ejpam-4854	153	10	n∑	n∑	PROPN
ejpam-4854	153	11	k=0	k=0	PROPN
ejpam-4854	153	12	〈	〈	PROPN
ejpam-4854	153	13	n	n	PROPN
ejpam-4854	153	14	k	k	X
ejpam-4854	153	15	〉	〉	X
ejpam-4854	153	16	β	β	X
ejpam-4854	153	17	,	,	PUNCT
ejpam-4854	153	18	r	r	NOUN
ejpam-4854	153	19	(	(	PUNCT
ejpam-4854	153	20	t−	t−	PRON
ejpam-4854	153	21	r)β	r)β	NOUN
ejpam-4854	153	22	,	,	PUNCT
ejpam-4854	153	23	k.	k.	PROPN
ejpam-4854	153	24	note	note	VERB
ejpam-4854	153	25	that	that	SCONJ
ejpam-4854	153	26	we	we	PRON
ejpam-4854	153	27	can	can	AUX
ejpam-4854	153	28	write	write	VERB
ejpam-4854	153	29	this	this	PRON
ejpam-4854	153	30	as	as	ADP
ejpam-4854	153	31	tn	tn	PROPN
ejpam-4854	153	32	=	=	SYM
ejpam-4854	153	33	n∑	n∑	PROPN
ejpam-4854	153	34	k=0	k=0	PROPN
ejpam-4854	153	35	〈	〈	PROPN
ejpam-4854	153	36	n	n	PROPN
ejpam-4854	153	37	k	k	X
ejpam-4854	153	38	〉	〉	PROPN
ejpam-4854	153	39	β	β	X
ejpam-4854	153	40	,	,	PUNCT
ejpam-4854	153	41	r	r	NOUN
ejpam-4854	153	42	(	(	PUNCT
ejpam-4854	153	43	t−r	t−r	PROPN
ejpam-4854	153	44	β	β	PROPN
ejpam-4854	153	45	k	k	NOUN
ejpam-4854	153	46	)	)	PUNCT
ejpam-4854	153	47	βkk	βkk	PROPN
ejpam-4854	153	48	!	!	PUNCT
ejpam-4854	153	49	.	.	PUNCT
ejpam-4854	154	1	replacing	replace	VERB
ejpam-4854	154	2	t	t	NOUN
ejpam-4854	154	3	by	by	ADP
ejpam-4854	154	4	βt+	βt+	ADJ
ejpam-4854	154	5	r	r	NOUN
ejpam-4854	154	6	,	,	PUNCT
ejpam-4854	154	7	we	we	PRON
ejpam-4854	154	8	have	have	VERB
ejpam-4854	154	9	(	(	PUNCT
ejpam-4854	154	10	βt+	βt+	ADJ
ejpam-4854	154	11	r)n	r)n	X
ejpam-4854	154	12	=	=	SYM
ejpam-4854	155	1	n∑	n∑	PROPN
ejpam-4854	155	2	k=0	k=0	PROPN
ejpam-4854	155	3	〈	〈	PROPN
ejpam-4854	155	4	n	n	PROPN
ejpam-4854	155	5	k	k	X
ejpam-4854	155	6	〉	〉	PROPN
ejpam-4854	155	7	β	β	X
ejpam-4854	155	8	,	,	PUNCT
ejpam-4854	155	9	r	r	NOUN
ejpam-4854	155	10	(	(	PUNCT
ejpam-4854	155	11	(	(	PUNCT
ejpam-4854	155	12	βt+r)−r	βt+r)−r	NOUN
ejpam-4854	155	13	β	β	X
ejpam-4854	155	14	k	k	NOUN
ejpam-4854	155	15	)	)	PUNCT
ejpam-4854	155	16	βkk	βkk	PROPN
ejpam-4854	155	17	!	!	PUNCT
ejpam-4854	156	1	(	(	PUNCT
ejpam-4854	156	2	βt+	βt+	ADJ
ejpam-4854	156	3	r)n	r)n	X
ejpam-4854	156	4	=	=	SYM
ejpam-4854	156	5	n∑	n∑	PROPN
ejpam-4854	156	6	k=0	k=0	PROPN
ejpam-4854	156	7	〈	〈	PROPN
ejpam-4854	156	8	n	n	PROPN
ejpam-4854	156	9	k	k	X
ejpam-4854	156	10	〉	〉	PROPN
ejpam-4854	156	11	β	β	X
ejpam-4854	156	12	,	,	PUNCT
ejpam-4854	156	13	r	r	NOUN
ejpam-4854	156	14	(	(	PUNCT
ejpam-4854	156	15	t	t	NOUN
ejpam-4854	156	16	k	k	PROPN
ejpam-4854	156	17	)	)	PUNCT
ejpam-4854	156	18	βkk	βkk	PROPN
ejpam-4854	156	19	!	!	PUNCT
ejpam-4854	156	20	.	.	PUNCT
ejpam-4854	157	1	(	(	PUNCT
ejpam-4854	157	2	6	6	X
ejpam-4854	157	3	)	)	PUNCT
ejpam-4854	157	4	this	this	DET
ejpam-4854	157	5	equation	equation	NOUN
ejpam-4854	157	6	(	(	PUNCT
ejpam-4854	157	7	6	6	NUM
ejpam-4854	157	8	)	)	PUNCT
ejpam-4854	157	9	can	can	AUX
ejpam-4854	157	10	be	be	AUX
ejpam-4854	157	11	represented	represent	VERB
ejpam-4854	157	12	by	by	ADP
ejpam-4854	157	13	the	the	DET
ejpam-4854	157	14	following	follow	VERB
ejpam-4854	157	15	system	system	NOUN
ejpam-4854	157	16	of	of	ADP
ejpam-4854	157	17	matrix	matrix	NOUN
ejpam-4854	157	18	equation	equation	NOUN
ejpam-4854	157	19	for	for	ADP
ejpam-4854	157	20	each	each	DET
ejpam-4854	157	21	n	n	NOUN
ejpam-4854	157	22	=	=	SYM
ejpam-4854	157	23	0	0	NUM
ejpam-4854	157	24	,	,	PUNCT
ejpam-4854	157	25	1	1	NUM
ejpam-4854	157	26	,	,	PUNCT
ejpam-4854	157	27	2	2	NUM
ejpam-4854	157	28	,	,	PUNCT
ejpam-4854	157	29	.	.	PUNCT
ejpam-4854	157	30	.	.	PUNCT
ejpam-4854	157	31	.	.	PUNCT
ejpam-4854	158	1	v(t	v(t	NUM
ejpam-4854	158	2	)	)	PUNCT
ejpam-4854	158	3	=	=	PUNCT
ejpam-4854	159	1	s̃r	s̃r	ADJ
ejpam-4854	159	2	,	,	PUNCT
ejpam-4854	159	3	β(n)cn(t	β(n)cn(t	NOUN
ejpam-4854	159	4	)	)	PUNCT
ejpam-4854	159	5	,	,	PUNCT
ejpam-4854	159	6	(	(	PUNCT
ejpam-4854	159	7	7	7	X
ejpam-4854	159	8	)	)	PUNCT
ejpam-4854	159	9	where	where	SCONJ
ejpam-4854	159	10	v(t	v(t	VERB
ejpam-4854	159	11	)	)	PUNCT
ejpam-4854	159	12	=	=	PUNCT
ejpam-4854	160	1	[	[	X
ejpam-4854	160	2	1	1	NUM
ejpam-4854	160	3	,	,	PUNCT
ejpam-4854	160	4	βt+	βt+	ADJ
ejpam-4854	160	5	r	r	NOUN
ejpam-4854	160	6	,	,	PUNCT
ejpam-4854	160	7	(	(	PUNCT
ejpam-4854	160	8	βt+	βt+	ADJ
ejpam-4854	160	9	r)2	r)2	NOUN
ejpam-4854	160	10	,	,	PUNCT
ejpam-4854	160	11	(	(	PUNCT
ejpam-4854	160	12	βt+	βt+	ADJ
ejpam-4854	160	13	r)3	r)3	NOUN
ejpam-4854	160	14	,	,	PUNCT
ejpam-4854	160	15	.	.	PUNCT
ejpam-4854	160	16	.	.	PUNCT
ejpam-4854	160	17	.	.	PUNCT
ejpam-4854	161	1	,	,	PUNCT
ejpam-4854	161	2	(	(	PUNCT
ejpam-4854	161	3	βt+	βt+	X
ejpam-4854	161	4	r)n−1	r)n−1	X
ejpam-4854	161	5	]	]	X
ejpam-4854	161	6	and	and	CCONJ
ejpam-4854	161	7	cn(t	cn(t	PUNCT
ejpam-4854	161	8	)	)	PUNCT
ejpam-4854	161	9	=	=	PUNCT
ejpam-4854	162	1	[	[	X
ejpam-4854	162	2	(	(	PUNCT
ejpam-4854	162	3	t	t	PROPN
ejpam-4854	162	4	0	0	NUM
ejpam-4854	162	5	)	)	PUNCT
ejpam-4854	162	6	,	,	PUNCT
ejpam-4854	162	7	β	β	X
ejpam-4854	162	8	(	(	PUNCT
ejpam-4854	162	9	t	t	PROPN
ejpam-4854	162	10	1	1	NUM
ejpam-4854	162	11	)	)	PUNCT
ejpam-4854	162	12	,	,	PUNCT
ejpam-4854	162	13	β2	β2	PROPN
ejpam-4854	162	14	(	(	PUNCT
ejpam-4854	162	15	t	t	PROPN
ejpam-4854	162	16	2	2	NUM
ejpam-4854	162	17	)	)	PUNCT
ejpam-4854	162	18	,	,	PUNCT
ejpam-4854	162	19	.	.	PUNCT
ejpam-4854	162	20	.	.	PUNCT
ejpam-4854	163	1	.	.	PUNCT
ejpam-4854	164	1	,	,	PUNCT
ejpam-4854	164	2	βn−1	βn−1	PROPN
ejpam-4854	164	3	(	(	PUNCT
ejpam-4854	164	4	t	t	PROPN
ejpam-4854	164	5	n−	n−	NOUN
ejpam-4854	164	6	1	1	NUM
ejpam-4854	164	7	)	)	PUNCT
ejpam-4854	164	8	]	]	PUNCT
ejpam-4854	164	9	g.	g.	PROPN
ejpam-4854	164	10	engalan	engalan	PROPN
ejpam-4854	164	11	,	,	PUNCT
ejpam-4854	164	12	m.r	m.r	PROPN
ejpam-4854	164	13	.	.	PROPN
ejpam-4854	164	14	latayada	latayada	PROPN
ejpam-4854	164	15	/	/	SYM
ejpam-4854	164	16	eur	eur	PROPN
ejpam-4854	164	17	.	.	PUNCT
ejpam-4854	165	1	j.	j.	PROPN
ejpam-4854	165	2	pure	pure	PROPN
ejpam-4854	165	3	appl	appl	PROPN
ejpam-4854	165	4	.	.	PROPN
ejpam-4854	165	5	math	math	PROPN
ejpam-4854	165	6	,	,	PUNCT
ejpam-4854	165	7	16	16	NUM
ejpam-4854	165	8	(	(	PUNCT
ejpam-4854	165	9	4	4	NUM
ejpam-4854	165	10	)	)	PUNCT
ejpam-4854	165	11	(	(	PUNCT
ejpam-4854	165	12	2023	2023	NUM
ejpam-4854	165	13	)	)	PUNCT
ejpam-4854	165	14	,	,	PUNCT
ejpam-4854	165	15	2306	2306	NUM
ejpam-4854	165	16	-	-	SYM
ejpam-4854	165	17	2322	2322	NUM
ejpam-4854	165	18	2314	2314	NUM
ejpam-4854	165	19	which	which	PRON
ejpam-4854	165	20	is	be	AUX
ejpam-4854	165	21	the	the	DET
ejpam-4854	165	22	first	first	ADJ
ejpam-4854	165	23	column	column	NOUN
ejpam-4854	165	24	of	of	ADP
ejpam-4854	165	25	the	the	DET
ejpam-4854	165	26	vβ	vβ	NOUN
ejpam-4854	165	27	,	,	PUNCT
ejpam-4854	165	28	r	r	NOUN
ejpam-4854	165	29	n	n	PROPN
ejpam-4854	165	30	(	(	PUNCT
ejpam-4854	165	31	t	t	PROPN
ejpam-4854	165	32	)	)	PUNCT
ejpam-4854	165	33	.	.	PUNCT
ejpam-4854	166	1	that	that	SCONJ
ejpam-4854	166	2	is,	is,	NOUN
ejpam-4854	166	3	1	1	NUM
ejpam-4854	166	4	βt+	βt+	NOUN
ejpam-4854	166	5	r	r	NOUN
ejpam-4854	166	6	(	(	PUNCT
ejpam-4854	166	7	βt+	βt+	ADJ
ejpam-4854	166	8	r)2	r)2	NOUN
ejpam-4854	166	9	(	(	PUNCT
ejpam-4854	166	10	βt+	βt+	ADJ
ejpam-4854	166	11	r)3	r)3	NOUN
ejpam-4854	166	12	...	...	PUNCT
ejpam-4854	166	13	(	(	PUNCT
ejpam-4854	166	14	βt+	βt+	ADJ
ejpam-4854	166	15	r)n−1	r)n−1	PROPN
ejpam-4854	166	16			NOUN
ejpam-4854	166	17	=	=	SYM
ejpam-4854	166	18			NOUN
ejpam-4854	166	19	1	1	NUM
ejpam-4854	166	20	0	0	NUM
ejpam-4854	166	21	0	0	NUM
ejpam-4854	166	22	0	0	NUM
ejpam-4854	166	23	0	0	NUM
ejpam-4854	167	1	r	r	NOUN
ejpam-4854	167	2	1	1	NUM
ejpam-4854	167	3	0	0	NUM
ejpam-4854	167	4	0	0	NUM
ejpam-4854	167	5	0	0	NUM
ejpam-4854	167	6	r2	r2	PROPN
ejpam-4854	167	7	β	β	X
ejpam-4854	167	8	+	+	CCONJ
ejpam-4854	167	9	2r	2r	NUM
ejpam-4854	167	10	2	2	NUM
ejpam-4854	167	11	0	0	NUM
ejpam-4854	167	12	0	0	NUM
ejpam-4854	167	13	...	...	PUNCT
ejpam-4854	167	14	...	...	PUNCT
ejpam-4854	167	15	...	...	PUNCT
ejpam-4854	167	16	.	.	PUNCT
ejpam-4854	167	17	.	.	PUNCT
ejpam-4854	167	18	.	.	PUNCT
ejpam-4854	168	1	...	...	PUNCT
ejpam-4854	169	1	rn−1	rn−1	PROPN
ejpam-4854	169	2	〈	〈	PROPN
ejpam-4854	169	3	n−	n−	NOUN
ejpam-4854	169	4	1	1	NUM
ejpam-4854	169	5	2	2	NUM
ejpam-4854	169	6	〉	〉	NOUN
ejpam-4854	169	7	β	β	NOUN
ejpam-4854	169	8	,	,	PUNCT
ejpam-4854	169	9	r	r	NOUN
ejpam-4854	169	10	〈	〈	PROPN
ejpam-4854	169	11	n−	n−	NOUN
ejpam-4854	169	12	1	1	NUM
ejpam-4854	169	13	3	3	NUM
ejpam-4854	169	14	〉	〉	NOUN
ejpam-4854	169	15	β	β	NOUN
ejpam-4854	169	16	,	,	PUNCT
ejpam-4854	169	17	r	r	NOUN
ejpam-4854	169	18	2	2	NUM
ejpam-4854	169	19	!	!	PUNCT
ejpam-4854	169	20	·	·	PUNCT
ejpam-4854	169	21	·	·	PUNCT
ejpam-4854	169	22	·	·	PUNCT
ejpam-4854	170	1	(	(	PUNCT
ejpam-4854	170	2	n−	n−	NOUN
ejpam-4854	170	3	1	1	NUM
ejpam-4854	170	4	)	)	PUNCT
ejpam-4854	170	5	!	!	PUNCT
ejpam-4854	171	1			VERB
ejpam-4854	171	2			NOUN
ejpam-4854	171	3	(	(	PUNCT
ejpam-4854	171	4	t	t	NOUN
ejpam-4854	171	5	0	0	NUM
ejpam-4854	171	6	)	)	PUNCT
ejpam-4854	172	1	β	β	PROPN
ejpam-4854	172	2	(	(	PUNCT
ejpam-4854	172	3	t	t	PROPN
ejpam-4854	172	4	1	1	NUM
ejpam-4854	172	5	)	)	PUNCT
ejpam-4854	172	6	β2	β2	NOUN
ejpam-4854	172	7	(	(	PUNCT
ejpam-4854	172	8	t	t	PROPN
ejpam-4854	172	9	2	2	NUM
ejpam-4854	172	10	)	)	PUNCT
ejpam-4854	172	11	...	...	PUNCT
ejpam-4854	173	1	βn−1	βn−1	PROPN
ejpam-4854	173	2	(	(	PUNCT
ejpam-4854	173	3	t	t	PROPN
ejpam-4854	173	4	n−1	n−1	PROPN
ejpam-4854	173	5	)	)	PUNCT
ejpam-4854	173	6			NOUN
ejpam-4854	173	7	.	.	PUNCT
ejpam-4854	174	1	thus	thus	ADV
ejpam-4854	174	2	,	,	PUNCT
ejpam-4854	174	3	by	by	ADP
ejpam-4854	174	4	equation	equation	NOUN
ejpam-4854	174	5	(	(	PUNCT
ejpam-4854	174	6	5	5	X
ejpam-4854	174	7	)	)	PUNCT
ejpam-4854	174	8	we	we	PRON
ejpam-4854	174	9	can	can	AUX
ejpam-4854	174	10	generalize	generalize	VERB
ejpam-4854	174	11	that	that	SCONJ
ejpam-4854	174	12	,	,	PUNCT
ejpam-4854	174	13	vβ	vβ	INTJ
ejpam-4854	174	14	,	,	PUNCT
ejpam-4854	174	15	r	r	NOUN
ejpam-4854	174	16	n	n	PROPN
ejpam-4854	174	17	(	(	PUNCT
ejpam-4854	174	18	t	t	PROPN
ejpam-4854	174	19	)	)	PUNCT
ejpam-4854	174	20	=	=	SYM
ejpam-4854	174	21	s̃(β	s̃(β	PROPN
ejpam-4854	174	22	,	,	PUNCT
ejpam-4854	174	23	r)(n)cβ	r)(n)cβ	NOUN
ejpam-4854	174	24	n	n	CCONJ
ejpam-4854	174	25	(	(	PUNCT
ejpam-4854	174	26	t	t	PROPN
ejpam-4854	174	27	)	)	PUNCT
ejpam-4854	174	28	.	.	PUNCT
ejpam-4854	175	1	example	example	NOUN
ejpam-4854	176	1	7	7	NUM
ejpam-4854	176	2	.	.	PUNCT
ejpam-4854	176	3	let	let	VERB
ejpam-4854	176	4	n	n	NOUN
ejpam-4854	176	5	=	=	SYM
ejpam-4854	176	6	4	4	X
ejpam-4854	176	7	.	.	PUNCT
ejpam-4854	177	1	then	then	ADV
ejpam-4854	177	2	we	we	PRON
ejpam-4854	177	3	have	have	VERB
ejpam-4854	177	4	matrices	matrix	NOUN
ejpam-4854	177	5	s̃(β	s̃(β	PROPN
ejpam-4854	177	6	,	,	PUNCT
ejpam-4854	177	7	r)(4	r)(4	NUM
ejpam-4854	177	8	)	)	PUNCT
ejpam-4854	178	1	=	=	SYM
ejpam-4854	178	2			NOUN
ejpam-4854	179	1	1	1	NUM
ejpam-4854	179	2	0	0	NUM
ejpam-4854	179	3	0	0	NUM
ejpam-4854	179	4	0	0	NUM
ejpam-4854	180	1	r	r	NOUN
ejpam-4854	180	2	1	1	NUM
ejpam-4854	180	3	0	0	NUM
ejpam-4854	180	4	0	0	NUM
ejpam-4854	180	5	r2	r2	NOUN
ejpam-4854	180	6	β	β	X
ejpam-4854	180	7	+	+	CCONJ
ejpam-4854	180	8	2r	2r	NUM
ejpam-4854	180	9	2	2	NUM
ejpam-4854	180	10	0	0	NUM
ejpam-4854	180	11	r3	r3	PROPN
ejpam-4854	180	12	β2	β2	NOUN
ejpam-4854	180	13	+	+	CCONJ
ejpam-4854	180	14	3βr	3βr	ADJ
ejpam-4854	180	15	+	+	CCONJ
ejpam-4854	180	16	3r2	3r2	NUM
ejpam-4854	180	17	6β	6β	NOUN
ejpam-4854	180	18	+	+	CCONJ
ejpam-4854	180	19	6r	6r	NUM
ejpam-4854	180	20	6	6	NUM
ejpam-4854	180	21			NOUN
ejpam-4854	180	22	and	and	CCONJ
ejpam-4854	180	23	cβ	cβ	NOUN
ejpam-4854	180	24	4	4	NUM
ejpam-4854	180	25	(	(	PUNCT
ejpam-4854	180	26	t	t	NOUN
ejpam-4854	180	27	)	)	PUNCT
ejpam-4854	180	28	=	=	PUNCT
ejpam-4854	180	29			NOUN
ejpam-4854	180	30	1	1	NUM
ejpam-4854	180	31	1	1	NUM
ejpam-4854	180	32	1	1	NUM
ejpam-4854	180	33	1	1	NUM
ejpam-4854	180	34	βt	βt	NOUN
ejpam-4854	180	35	β(t+	β(t+	NOUN
ejpam-4854	180	36	1	1	NUM
ejpam-4854	180	37	)	)	PUNCT
ejpam-4854	180	38	β(t+	β(t+	NOUN
ejpam-4854	180	39	2	2	NUM
ejpam-4854	180	40	)	)	PUNCT
ejpam-4854	180	41	β(t+	β(t+	NOUN
ejpam-4854	180	42	3	3	NUM
ejpam-4854	180	43	)	)	PUNCT
ejpam-4854	180	44	β2t(t−1	β2t(t−1	PROPN
ejpam-4854	180	45	)	)	PUNCT
ejpam-4854	180	46	2	2	NUM
ejpam-4854	180	47	β2(t+1)t	β2(t+1)t	SYM
ejpam-4854	180	48	2	2	NUM
ejpam-4854	180	49	β2(t+2)(t+1	β2(t+2)(t+1	SYM
ejpam-4854	180	50	)	)	PUNCT
ejpam-4854	180	51	2	2	NUM
ejpam-4854	180	52	β2(t+3)(t+2	β2(t+3)(t+2	NOUN
ejpam-4854	180	53	)	)	PUNCT
ejpam-4854	180	54	2	2	NUM
ejpam-4854	180	55	β3t(t−1)(t−2	β3t(t−1)(t−2	NOUN
ejpam-4854	180	56	)	)	PUNCT
ejpam-4854	180	57	6	6	NUM
ejpam-4854	180	58	β3(t+1)t(t−1	β3(t+1)t(t−1	NOUN
ejpam-4854	180	59	)	)	PUNCT
ejpam-4854	180	60	6	6	NUM
ejpam-4854	180	61	β3(t+2)(t+1)(t	β3(t+2)(t+1)(t	NOUN
ejpam-4854	180	62	)	)	PUNCT
ejpam-4854	180	63	6	6	NUM
ejpam-4854	180	64	β3(t+3)(t+2)(t+1	β3(t+3)(t+2)(t+1	SYM
ejpam-4854	180	65	)	)	PUNCT
ejpam-4854	180	66	6	6	NUM
ejpam-4854	180	67			ADJ
ejpam-4854	180	68	.	.	PUNCT
ejpam-4854	181	1	now	now	ADV
ejpam-4854	181	2	,	,	PUNCT
ejpam-4854	181	3	s̃(β	s̃(β	PROPN
ejpam-4854	181	4	,	,	PUNCT
ejpam-4854	181	5	r)(4)cβ	r)(4)cβ	NOUN
ejpam-4854	181	6	4	4	NUM
ejpam-4854	181	7	(	(	PUNCT
ejpam-4854	181	8	t	t	NOUN
ejpam-4854	181	9	)	)	PUNCT
ejpam-4854	181	10	=	=	SYM
ejpam-4854	181	11			NOUN
ejpam-4854	181	12	1	1	NUM
ejpam-4854	181	13	0	0	NUM
ejpam-4854	181	14	0	0	NUM
ejpam-4854	181	15	0	0	NUM
ejpam-4854	182	1	r	r	NOUN
ejpam-4854	182	2	1	1	NUM
ejpam-4854	182	3	0	0	NUM
ejpam-4854	182	4	0	0	NUM
ejpam-4854	182	5	r2	r2	NOUN
ejpam-4854	182	6	β	β	X
ejpam-4854	182	7	+	+	CCONJ
ejpam-4854	182	8	2r	2r	NUM
ejpam-4854	182	9	2	2	NUM
ejpam-4854	182	10	0	0	NUM
ejpam-4854	182	11	r3	r3	PROPN
ejpam-4854	182	12	β2	β2	NOUN
ejpam-4854	182	13	+	+	CCONJ
ejpam-4854	182	14	3βr	3βr	ADJ
ejpam-4854	182	15	+	+	CCONJ
ejpam-4854	182	16	3r2	3r2	NUM
ejpam-4854	182	17	6β	6β	NOUN
ejpam-4854	182	18	+	+	CCONJ
ejpam-4854	182	19	6r	6r	NUM
ejpam-4854	182	20	6	6	NUM
ejpam-4854	182	21			ADJ
ejpam-4854	182	22	·	·	PUNCT
ejpam-4854	182	23			NOUN
ejpam-4854	182	24	1	1	NUM
ejpam-4854	182	25	1	1	NUM
ejpam-4854	182	26	1	1	NUM
ejpam-4854	182	27	1	1	NUM
ejpam-4854	182	28	βt	βt	NOUN
ejpam-4854	182	29	β(t+	β(t+	NOUN
ejpam-4854	182	30	1	1	NUM
ejpam-4854	182	31	)	)	PUNCT
ejpam-4854	182	32	β(t+	β(t+	NOUN
ejpam-4854	182	33	2	2	NUM
ejpam-4854	182	34	)	)	PUNCT
ejpam-4854	182	35	β(t+	β(t+	NOUN
ejpam-4854	182	36	3	3	NUM
ejpam-4854	182	37	)	)	PUNCT
ejpam-4854	182	38	β2t(t−1	β2t(t−1	PROPN
ejpam-4854	182	39	)	)	PUNCT
ejpam-4854	182	40	2	2	NUM
ejpam-4854	182	41	β2(t+1)t	β2(t+1)t	SYM
ejpam-4854	182	42	2	2	NUM
ejpam-4854	182	43	β2(t+2)(t+1	β2(t+2)(t+1	SYM
ejpam-4854	182	44	)	)	PUNCT
ejpam-4854	182	45	2	2	NUM
ejpam-4854	182	46	β2(t+3)(t+2	β2(t+3)(t+2	NOUN
ejpam-4854	182	47	)	)	PUNCT
ejpam-4854	182	48	2	2	NUM
ejpam-4854	182	49	β3t(t−1)(t−2	β3t(t−1)(t−2	NOUN
ejpam-4854	182	50	)	)	PUNCT
ejpam-4854	182	51	6	6	NUM
ejpam-4854	182	52	β3(t+1)t(t−1	β3(t+1)t(t−1	NOUN
ejpam-4854	182	53	)	)	PUNCT
ejpam-4854	182	54	6	6	NUM
ejpam-4854	182	55	β3(t+2)(t+1)(t	β3(t+2)(t+1)(t	NOUN
ejpam-4854	182	56	)	)	PUNCT
ejpam-4854	182	57	6	6	NUM
ejpam-4854	182	58	β3(t+3)(t+2)(t+1	β3(t+3)(t+2)(t+1	SYM
ejpam-4854	182	59	)	)	PUNCT
ejpam-4854	182	60	6	6	NUM
ejpam-4854	182	61			VERB
ejpam-4854	182	62	=	=	NOUN
ejpam-4854	182	63			NOUN
ejpam-4854	182	64	1	1	NUM
ejpam-4854	182	65	1	1	NUM
ejpam-4854	182	66	1	1	NUM
ejpam-4854	182	67	1	1	NUM
ejpam-4854	182	68	βt+	βt+	NOUN
ejpam-4854	182	69	r	r	NOUN
ejpam-4854	182	70	βt+	βt+	X
ejpam-4854	183	1	β	β	X
ejpam-4854	183	2	+	+	NOUN
ejpam-4854	183	3	r	r	NOUN
ejpam-4854	183	4	βt+	βt+	ADJ
ejpam-4854	183	5	2β	2β	NOUN
ejpam-4854	184	1	+	+	CCONJ
ejpam-4854	184	2	r	r	NOUN
ejpam-4854	184	3	βt+	βt+	ADJ
ejpam-4854	184	4	3β	3β	NOUN
ejpam-4854	184	5	+	+	CCONJ
ejpam-4854	185	1	r	r	NOUN
ejpam-4854	185	2	(	(	PUNCT
ejpam-4854	185	3	βt+	βt+	ADJ
ejpam-4854	185	4	r)2	r)2	NOUN
ejpam-4854	185	5	(	(	PUNCT
ejpam-4854	185	6	βt+	βt+	ADJ
ejpam-4854	185	7	β	β	X
ejpam-4854	185	8	+	+	ADJ
ejpam-4854	185	9	r)2	r)2	NOUN
ejpam-4854	185	10	(	(	PUNCT
ejpam-4854	185	11	βt+	βt+	ADJ
ejpam-4854	185	12	2β	2β	NOUN
ejpam-4854	185	13	+	+	CCONJ
ejpam-4854	185	14	r)2	r)2	NOUN
ejpam-4854	185	15	(	(	PUNCT
ejpam-4854	185	16	βt+	βt+	ADJ
ejpam-4854	185	17	3β	3β	NOUN
ejpam-4854	185	18	+	+	CCONJ
ejpam-4854	185	19	r)2	r)2	NOUN
ejpam-4854	185	20	(	(	PUNCT
ejpam-4854	185	21	βt+	βt+	ADJ
ejpam-4854	185	22	r)3	r)3	NOUN
ejpam-4854	185	23	(	(	PUNCT
ejpam-4854	185	24	βt+	βt+	ADJ
ejpam-4854	185	25	β	β	X
ejpam-4854	185	26	+	+	CCONJ
ejpam-4854	185	27	r)3	r)3	NOUN
ejpam-4854	185	28	(	(	PUNCT
ejpam-4854	185	29	βt+	βt+	ADJ
ejpam-4854	185	30	2β	2β	NOUN
ejpam-4854	185	31	+	+	CCONJ
ejpam-4854	185	32	r)3	r)3	PROPN
ejpam-4854	185	33	(	(	PUNCT
ejpam-4854	185	34	βt+	βt+	ADJ
ejpam-4854	185	35	3β	3β	NOUN
ejpam-4854	185	36	+	+	CCONJ
ejpam-4854	185	37	r)3	r)3	NOUN
ejpam-4854	185	38			NOUN
ejpam-4854	185	39	=	=	SYM
ejpam-4854	185	40	vβ	vβ	NOUN
ejpam-4854	185	41	,	,	PUNCT
ejpam-4854	185	42	r	r	NOUN
ejpam-4854	185	43	4	4	NUM
ejpam-4854	185	44	(	(	PUNCT
ejpam-4854	185	45	t	t	PROPN
ejpam-4854	185	46	)	)	PUNCT
ejpam-4854	185	47	g.	g.	PROPN
ejpam-4854	185	48	engalan	engalan	PROPN
ejpam-4854	185	49	,	,	PUNCT
ejpam-4854	185	50	m.r	m.r	PROPN
ejpam-4854	185	51	.	.	PROPN
ejpam-4854	185	52	latayada	latayada	PROPN
ejpam-4854	185	53	/	/	SYM
ejpam-4854	185	54	eur	eur	PROPN
ejpam-4854	185	55	.	.	PUNCT
ejpam-4854	186	1	j.	j.	PROPN
ejpam-4854	186	2	pure	pure	PROPN
ejpam-4854	186	3	appl	appl	PROPN
ejpam-4854	186	4	.	.	PROPN
ejpam-4854	186	5	math	math	PROPN
ejpam-4854	186	6	,	,	PUNCT
ejpam-4854	186	7	16	16	NUM
ejpam-4854	186	8	(	(	PUNCT
ejpam-4854	186	9	4	4	NUM
ejpam-4854	186	10	)	)	PUNCT
ejpam-4854	186	11	(	(	PUNCT
ejpam-4854	186	12	2023	2023	NUM
ejpam-4854	186	13	)	)	PUNCT
ejpam-4854	186	14	,	,	PUNCT
ejpam-4854	186	15	2306	2306	NUM
ejpam-4854	186	16	-	-	SYM
ejpam-4854	186	17	2322	2322	NUM
ejpam-4854	186	18	2315	2315	NUM
ejpam-4854	186	19	corollary	corollary	NOUN
ejpam-4854	186	20	1	1	NUM
ejpam-4854	186	21	.	.	PUNCT
ejpam-4854	187	1	for	for	ADP
ejpam-4854	187	2	any	any	DET
ejpam-4854	187	3	real	real	ADJ
ejpam-4854	187	4	number	number	NOUN
ejpam-4854	187	5	t	t	PROPN
ejpam-4854	187	6	,	,	PUNCT
ejpam-4854	187	7	we	we	PRON
ejpam-4854	187	8	have	have	VERB
ejpam-4854	187	9	vβ	vβ	ADJ
ejpam-4854	187	10	,	,	PUNCT
ejpam-4854	187	11	r	r	NOUN
ejpam-4854	187	12	n	n	PROPN
ejpam-4854	187	13	(	(	PUNCT
ejpam-4854	187	14	t	t	PROPN
ejpam-4854	187	15	)	)	PUNCT
ejpam-4854	187	16	=	=	SYM
ejpam-4854	188	1	s̃(β	s̃(β	PROPN
ejpam-4854	188	2	,	,	PUNCT
ejpam-4854	188	3	r)(n)	r)(n)	X
ejpam-4854	188	4	△	△	NOUN
ejpam-4854	188	5	n(t)p	n(t)p	PROPN
ejpam-4854	188	6	t	t	NOUN
ejpam-4854	188	7	n	n	PRON
ejpam-4854	188	8	where	where	SCONJ
ejpam-4854	188	9	[	[	X
ejpam-4854	188	10	△	△	NOUN
ejpam-4854	188	11	n(t)]ij	n(t)]ij	ADJ
ejpam-4854	188	12	=	=	SYM
ejpam-4854	188	13	βi	βi	PROPN
ejpam-4854	188	14	(	(	PUNCT
ejpam-4854	188	15	t	t	PROPN
ejpam-4854	188	16	i−	i−	PROPN
ejpam-4854	188	17	j	j	PROPN
ejpam-4854	188	18	)	)	PUNCT
ejpam-4854	188	19	and	and	CCONJ
ejpam-4854	188	20	p	p	PROPN
ejpam-4854	188	21	t	t	PROPN
ejpam-4854	188	22	n	n	X
ejpam-4854	188	23	is	be	AUX
ejpam-4854	188	24	the	the	DET
ejpam-4854	188	25	transpose	transpose	NOUN
ejpam-4854	188	26	of	of	ADP
ejpam-4854	188	27	the	the	DET
ejpam-4854	188	28	pascal	pascal	ADJ
ejpam-4854	188	29	matrix	matrix	NOUN
ejpam-4854	188	30	pn	pn	NOUN
ejpam-4854	188	31	proof	proof	NOUN
ejpam-4854	188	32	.	.	PUNCT
ejpam-4854	189	1	from	from	ADP
ejpam-4854	189	2	vandermonde	vandermonde	PROPN
ejpam-4854	189	3	’s	’s	PART
ejpam-4854	189	4	convolution	convolution	NOUN
ejpam-4854	189	5	identity	identity	NOUN
ejpam-4854	189	6	,	,	PUNCT
ejpam-4854	189	7	(	(	PUNCT
ejpam-4854	189	8	m+	m+	NOUN
ejpam-4854	189	9	n	n	NOUN
ejpam-4854	189	10	r	r	NOUN
ejpam-4854	189	11	)	)	PUNCT
ejpam-4854	189	12	=	=	SYM
ejpam-4854	189	13	(	(	PUNCT
ejpam-4854	189	14	mi	mi	PROPN
ejpam-4854	189	15	)	)	PUNCT
ejpam-4854	189	16	(	(	PUNCT
ejpam-4854	189	17	n	n	X
ejpam-4854	189	18	r−i),∑	r−i),∑	PRON
ejpam-4854	189	19	i=0	i=0	PROPN
ejpam-4854	189	20	we	we	PRON
ejpam-4854	189	21	can	can	AUX
ejpam-4854	189	22	obtain	obtain	VERB
ejpam-4854	189	23	the	the	DET
ejpam-4854	189	24	lu	lu	NOUN
ejpam-4854	189	25	factorization	factorization	NOUN
ejpam-4854	189	26	of	of	ADP
ejpam-4854	189	27	cβ	cβ	NOUN
ejpam-4854	189	28	n	n	PROPN
ejpam-4854	189	29	(	(	PUNCT
ejpam-4854	189	30	t	t	PROPN
ejpam-4854	189	31	)	)	PUNCT
ejpam-4854	189	32	.	.	PUNCT
ejpam-4854	190	1	that	that	PRON
ejpam-4854	190	2	is	be	AUX
ejpam-4854	190	3	,	,	PUNCT
ejpam-4854	190	4	cβ	cβ	NOUN
ejpam-4854	190	5	n	n	PROPN
ejpam-4854	190	6	(	(	PUNCT
ejpam-4854	190	7	t	t	PROPN
ejpam-4854	190	8	)	)	PUNCT
ejpam-4854	190	9	=	=	PUNCT
ejpam-4854	191	1	[	[	PUNCT
ejpam-4854	191	2	βi	βi	X
ejpam-4854	191	3	(	(	PUNCT
ejpam-4854	191	4	t+	t+	NOUN
ejpam-4854	191	5	j	j	PROPN
ejpam-4854	191	6	i	i	PROPN
ejpam-4854	191	7	)	)	PUNCT
ejpam-4854	191	8	]	]	PUNCT
ejpam-4854	191	9	0≤i	0≤i	PROPN
ejpam-4854	191	10	,	,	PUNCT
ejpam-4854	191	11	j≤n−1	j≤n−1	PROPN
ejpam-4854	192	1	=	=	PUNCT
ejpam-4854	193	1	[	[	PUNCT
ejpam-4854	193	2	βi	βi	X
ejpam-4854	193	3	i∑	i∑	PROPN
ejpam-4854	193	4	k=0	k=0	PROPN
ejpam-4854	193	5	(	(	PUNCT
ejpam-4854	193	6	j	j	NOUN
ejpam-4854	193	7	i	i	PROPN
ejpam-4854	193	8	)	)	PUNCT
ejpam-4854	193	9	(	(	PUNCT
ejpam-4854	193	10	t	t	PROPN
ejpam-4854	193	11	i−	i−	PROPN
ejpam-4854	193	12	k	k	PROPN
ejpam-4854	193	13	)	)	PUNCT
ejpam-4854	193	14	]	]	PUNCT
ejpam-4854	194	1	0≤i	0≤i	PROPN
ejpam-4854	194	2	,	,	PUNCT
ejpam-4854	194	3	j≤n−1	j≤n−1	PROPN
ejpam-4854	194	4	=	=	PUNCT
ejpam-4854	195	1	[	[	PUNCT
ejpam-4854	195	2	i∑	i∑	PROPN
ejpam-4854	195	3	k=0	k=0	PROPN
ejpam-4854	195	4	βi	βi	PROPN
ejpam-4854	195	5	(	(	PUNCT
ejpam-4854	195	6	j	j	PROPN
ejpam-4854	195	7	i	i	PROPN
ejpam-4854	195	8	)	)	PUNCT
ejpam-4854	195	9	(	(	PUNCT
ejpam-4854	195	10	t	t	PROPN
ejpam-4854	195	11	i−	i−	PROPN
ejpam-4854	195	12	k	k	PROPN
ejpam-4854	195	13	)	)	PUNCT
ejpam-4854	195	14	]	]	PUNCT
ejpam-4854	196	1	0≤i	0≤i	PROPN
ejpam-4854	196	2	,	,	PUNCT
ejpam-4854	196	3	j≤n−1	j≤n−1	PROPN
ejpam-4854	196	4	let	let	VERB
ejpam-4854	196	5	[	[	X
ejpam-4854	196	6	△	△	NOUN
ejpam-4854	196	7	n(t)]ij	n(t)]ij	ADJ
ejpam-4854	196	8	=	=	SYM
ejpam-4854	196	9	βi	βi	PROPN
ejpam-4854	196	10	(	(	PUNCT
ejpam-4854	196	11	t	t	PROPN
ejpam-4854	196	12	i−	i−	PROPN
ejpam-4854	196	13	j	j	PROPN
ejpam-4854	196	14	)	)	PUNCT
ejpam-4854	196	15	.	.	PUNCT
ejpam-4854	197	1	note	note	VERB
ejpam-4854	197	2	the	the	DET
ejpam-4854	197	3	[	[	X
ejpam-4854	197	4	(	(	PUNCT
ejpam-4854	197	5	j	j	PROPN
ejpam-4854	197	6	i	i	PROPN
ejpam-4854	197	7	)	)	PUNCT
ejpam-4854	197	8	]	]	PUNCT
ejpam-4854	198	1	ij	ij	NOUN
ejpam-4854	198	2	is	be	AUX
ejpam-4854	198	3	the	the	DET
ejpam-4854	198	4	transpose	transpose	NOUN
ejpam-4854	198	5	of	of	ADP
ejpam-4854	198	6	the	the	DET
ejpam-4854	198	7	pascal	pascal	ADJ
ejpam-4854	198	8	matrix	matrix	NOUN
ejpam-4854	198	9	pn	pn	NOUN
ejpam-4854	198	10	=	=	SYM
ejpam-4854	198	11	(	(	PUNCT
ejpam-4854	198	12	(	(	PUNCT
ejpam-4854	198	13	i	i	PRON
ejpam-4854	198	14	j	j	PROPN
ejpam-4854	198	15	)	)	PUNCT
ejpam-4854	198	16	)	)	PUNCT
ejpam-4854	199	1	ij	ij	INTJ
ejpam-4854	199	2	.	.	PUNCT
ejpam-4854	200	1	then	then	ADV
ejpam-4854	200	2	we	we	PRON
ejpam-4854	200	3	can	can	AUX
ejpam-4854	200	4	write	write	VERB
ejpam-4854	200	5	cβ	cβ	PROPN
ejpam-4854	200	6	n	n	PROPN
ejpam-4854	200	7	(	(	PUNCT
ejpam-4854	200	8	t	t	PROPN
ejpam-4854	200	9	)	)	PUNCT
ejpam-4854	200	10	as	as	ADP
ejpam-4854	200	11	cβ	cβ	PROPN
ejpam-4854	200	12	n	n	PROPN
ejpam-4854	200	13	(	(	PUNCT
ejpam-4854	200	14	t	t	PROPN
ejpam-4854	200	15	)	)	PUNCT
ejpam-4854	200	16	=	=	PUNCT
ejpam-4854	201	1	△	△	PROPN
ejpam-4854	201	2	n(t)p	n(t)p	PROPN
ejpam-4854	201	3	t	t	PROPN
ejpam-4854	201	4	n	n	ADV
ejpam-4854	201	5	.	.	PUNCT
ejpam-4854	202	1	by	by	ADP
ejpam-4854	202	2	theorem	theorem	NOUN
ejpam-4854	202	3	1	1	NUM
ejpam-4854	202	4	,	,	PUNCT
ejpam-4854	202	5	vβ	vβ	NOUN
ejpam-4854	202	6	,	,	PUNCT
ejpam-4854	202	7	r	r	NOUN
ejpam-4854	202	8	n	n	PROPN
ejpam-4854	202	9	(	(	PUNCT
ejpam-4854	202	10	t	t	PROPN
ejpam-4854	202	11	)	)	PUNCT
ejpam-4854	202	12	=	=	SYM
ejpam-4854	202	13	s̃(β	s̃(β	PROPN
ejpam-4854	202	14	,	,	PUNCT
ejpam-4854	202	15	r)(n)	r)(n)	X
ejpam-4854	202	16	△	△	NOUN
ejpam-4854	202	17	n(t)p	n(t)p	PROPN
ejpam-4854	202	18	t	t	PROPN
ejpam-4854	202	19	n	n	NOUN
ejpam-4854	202	20	.	.	PUNCT
ejpam-4854	202	21	example	example	NOUN
ejpam-4854	203	1	8	8	NUM
ejpam-4854	203	2	.	.	PUNCT
ejpam-4854	204	1	let	let	VERB
ejpam-4854	204	2	t	t	NOUN
ejpam-4854	204	3	=	=	SYM
ejpam-4854	204	4	1	1	NUM
ejpam-4854	204	5	,	,	PUNCT
ejpam-4854	204	6	and	and	CCONJ
ejpam-4854	204	7	n	n	CCONJ
ejpam-4854	204	8	=	=	SYM
ejpam-4854	205	1	4	4	NUM
ejpam-4854	205	2	.	.	PUNCT
ejpam-4854	206	1	then	then	ADV
ejpam-4854	206	2	,	,	PUNCT
ejpam-4854	206	3	s̃(β	s̃(β	PROPN
ejpam-4854	206	4	,	,	PUNCT
ejpam-4854	206	5	r)(4	r)(4	NUM
ejpam-4854	206	6	)	)	PUNCT
ejpam-4854	207	1	=	=	SYM
ejpam-4854	207	2			NOUN
ejpam-4854	208	1	1	1	NUM
ejpam-4854	208	2	0	0	NUM
ejpam-4854	208	3	0	0	NUM
ejpam-4854	208	4	0	0	NUM
ejpam-4854	209	1	r	r	NOUN
ejpam-4854	209	2	1	1	NUM
ejpam-4854	209	3	0	0	NUM
ejpam-4854	209	4	0	0	NUM
ejpam-4854	209	5	r2	r2	NOUN
ejpam-4854	209	6	β	β	X
ejpam-4854	209	7	+	+	CCONJ
ejpam-4854	209	8	2r	2r	NUM
ejpam-4854	209	9	2	2	NUM
ejpam-4854	209	10	0	0	NUM
ejpam-4854	209	11	r3	r3	NOUN
ejpam-4854	209	12	β3	β3	NOUN
ejpam-4854	209	13	+	+	CCONJ
ejpam-4854	209	14	3βr	3βr	ADJ
ejpam-4854	209	15	+	+	CCONJ
ejpam-4854	209	16	r2	r2	NOUN
ejpam-4854	209	17	6β	6β	NOUN
ejpam-4854	209	18	+	+	CCONJ
ejpam-4854	209	19	6r	6r	NUM
ejpam-4854	209	20	6	6	NUM
ejpam-4854	209	21			NOUN
ejpam-4854	209	22	,	,	PUNCT
ejpam-4854	209	23	△	△	NOUN
ejpam-4854	209	24	4(t	4(t	NUM
ejpam-4854	209	25	)	)	PUNCT
ejpam-4854	209	26	=	=	SYM
ejpam-4854	209	27			NOUN
ejpam-4854	210	1	1	1	NUM
ejpam-4854	210	2	0	0	NUM
ejpam-4854	210	3	0	0	NUM
ejpam-4854	210	4	0	0	NUM
ejpam-4854	210	5	β	β	X
ejpam-4854	210	6	beta	beta	NOUN
ejpam-4854	210	7	0	0	NUM
ejpam-4854	210	8	0	0	NUM
ejpam-4854	210	9	0	0	NUM
ejpam-4854	210	10	β2	β2	NOUN
ejpam-4854	210	11	β2	β2	NOUN
ejpam-4854	210	12	0	0	NUM
ejpam-4854	210	13	0	0	NUM
ejpam-4854	210	14	0	0	NUM
ejpam-4854	210	15	β3	β3	NOUN
ejpam-4854	210	16	β3	β3	VERB
ejpam-4854	210	17	0	0	NUM
ejpam-4854	210	18			PROPN
ejpam-4854	210	19	g.	g.	PROPN
ejpam-4854	210	20	engalan	engalan	PROPN
ejpam-4854	210	21	,	,	PUNCT
ejpam-4854	210	22	m.r	m.r	PROPN
ejpam-4854	210	23	.	.	PROPN
ejpam-4854	210	24	latayada	latayada	PROPN
ejpam-4854	210	25	/	/	SYM
ejpam-4854	210	26	eur	eur	PROPN
ejpam-4854	210	27	.	.	PUNCT
ejpam-4854	211	1	j.	j.	PROPN
ejpam-4854	211	2	pure	pure	PROPN
ejpam-4854	211	3	appl	appl	PROPN
ejpam-4854	211	4	.	.	PROPN
ejpam-4854	211	5	math	math	PROPN
ejpam-4854	211	6	,	,	PUNCT
ejpam-4854	211	7	16	16	NUM
ejpam-4854	211	8	(	(	PUNCT
ejpam-4854	211	9	4	4	NUM
ejpam-4854	211	10	)	)	PUNCT
ejpam-4854	211	11	(	(	PUNCT
ejpam-4854	211	12	2023	2023	NUM
ejpam-4854	211	13	)	)	PUNCT
ejpam-4854	211	14	,	,	PUNCT
ejpam-4854	211	15	2306	2306	NUM
ejpam-4854	211	16	-	-	SYM
ejpam-4854	211	17	2322	2322	NUM
ejpam-4854	211	18	2316	2316	NUM
ejpam-4854	211	19	and	and	CCONJ
ejpam-4854	211	20	p	p	NOUN
ejpam-4854	211	21	t	t	PROPN
ejpam-4854	211	22	4	4	NUM
ejpam-4854	211	23	=	=	SYM
ejpam-4854	211	24			NOUN
ejpam-4854	211	25	1	1	NUM
ejpam-4854	211	26	1	1	NUM
ejpam-4854	211	27	1	1	NUM
ejpam-4854	211	28	1	1	NUM
ejpam-4854	211	29	0	0	NUM
ejpam-4854	211	30	1	1	NUM
ejpam-4854	211	31	2	2	NUM
ejpam-4854	211	32	3	3	NUM
ejpam-4854	211	33	0	0	NUM
ejpam-4854	211	34	0	0	NUM
ejpam-4854	211	35	1	1	NUM
ejpam-4854	211	36	3	3	NUM
ejpam-4854	211	37	0	0	NUM
ejpam-4854	211	38	0	0	NUM
ejpam-4854	211	39	0	0	NUM
ejpam-4854	211	40	1	1	NUM
ejpam-4854	211	41			NOUN
ejpam-4854	211	42	.	.	PUNCT
ejpam-4854	212	1	now	now	ADV
ejpam-4854	212	2	,	,	PUNCT
ejpam-4854	212	3	s̃(β	s̃(β	PROPN
ejpam-4854	212	4	,	,	PUNCT
ejpam-4854	212	5	r)(n)	r)(n)	X
ejpam-4854	212	6	△	△	NOUN
ejpam-4854	212	7	n(t)p	n(t)p	PROPN
ejpam-4854	212	8	t	t	PROPN
ejpam-4854	212	9	n	n	NOUN
ejpam-4854	212	10	=	=	SYM
ejpam-4854	212	11			NOUN
ejpam-4854	213	1	1	1	NUM
ejpam-4854	213	2	0	0	NUM
ejpam-4854	213	3	0	0	NUM
ejpam-4854	213	4	0	0	NUM
ejpam-4854	214	1	r	r	NOUN
ejpam-4854	214	2	1	1	NUM
ejpam-4854	214	3	0	0	NUM
ejpam-4854	214	4	0	0	NUM
ejpam-4854	214	5	r2	r2	NOUN
ejpam-4854	214	6	β	β	X
ejpam-4854	214	7	+	+	CCONJ
ejpam-4854	214	8	2r	2r	NUM
ejpam-4854	214	9	2	2	NUM
ejpam-4854	214	10	0	0	NUM
ejpam-4854	214	11	r3	r3	NOUN
ejpam-4854	214	12	β3	β3	NOUN
ejpam-4854	214	13	+	+	CCONJ
ejpam-4854	214	14	3βr	3βr	ADJ
ejpam-4854	214	15	+	+	CCONJ
ejpam-4854	214	16	r2	r2	NOUN
ejpam-4854	214	17	6β	6β	NOUN
ejpam-4854	214	18	+	+	CCONJ
ejpam-4854	214	19	6r	6r	NUM
ejpam-4854	214	20	6	6	NUM
ejpam-4854	214	21			NOUN
ejpam-4854	214	22	×	×	NOUN
ejpam-4854	214	23			NOUN
ejpam-4854	214	24	1	1	NUM
ejpam-4854	214	25	0	0	NUM
ejpam-4854	214	26	0	0	NUM
ejpam-4854	214	27	0	0	NUM
ejpam-4854	214	28	β	β	X
ejpam-4854	214	29	β	β	X
ejpam-4854	214	30	0	0	NUM
ejpam-4854	214	31	0	0	SYM
ejpam-4854	214	32	0	0	NUM
ejpam-4854	214	33	β2	β2	NOUN
ejpam-4854	214	34	β2	β2	NOUN
ejpam-4854	214	35	0	0	NUM
ejpam-4854	214	36	0	0	NUM
ejpam-4854	214	37	0	0	NUM
ejpam-4854	214	38	β3	β3	NOUN
ejpam-4854	214	39	β3	β3	VERB
ejpam-4854	214	40			NOUN
ejpam-4854	214	41			NOUN
ejpam-4854	214	42	1	1	NUM
ejpam-4854	214	43	1	1	NUM
ejpam-4854	214	44	1	1	NUM
ejpam-4854	214	45	1	1	NUM
ejpam-4854	214	46	0	0	NUM
ejpam-4854	214	47	1	1	NUM
ejpam-4854	214	48	2	2	NUM
ejpam-4854	214	49	3	3	NUM
ejpam-4854	214	50	0	0	NUM
ejpam-4854	214	51	0	0	NUM
ejpam-4854	214	52	1	1	NUM
ejpam-4854	214	53	3	3	NUM
ejpam-4854	214	54	0	0	NUM
ejpam-4854	214	55	0	0	NUM
ejpam-4854	214	56	0	0	NUM
ejpam-4854	214	57	1	1	NUM
ejpam-4854	214	58			NOUN
ejpam-4854	214	59	=	=	PUNCT
ejpam-4854	214	60			NOUN
ejpam-4854	214	61	1	1	NUM
ejpam-4854	214	62	0	0	NUM
ejpam-4854	214	63	0	0	NUM
ejpam-4854	214	64	0	0	NUM
ejpam-4854	215	1	β	β	X
ejpam-4854	216	1	+	+	X
ejpam-4854	216	2	r	r	NOUN
ejpam-4854	216	3	β	β	NOUN
ejpam-4854	216	4	0	0	NUM
ejpam-4854	216	5	0	0	NUM
ejpam-4854	217	1	(	(	PUNCT
ejpam-4854	217	2	β	β	X
ejpam-4854	217	3	+	+	PUNCT
ejpam-4854	217	4	r)2	r)2	NOUN
ejpam-4854	217	5	β(3β	β(3β	PUNCT
ejpam-4854	218	1	+	+	NUM
ejpam-4854	218	2	2r	2r	NUM
ejpam-4854	218	3	)	)	PUNCT
ejpam-4854	218	4	2β2	2β2	NUM
ejpam-4854	218	5	0	0	NUM
ejpam-4854	218	6	(	(	PUNCT
ejpam-4854	218	7	β	β	X
ejpam-4854	218	8	+	+	CCONJ
ejpam-4854	218	9	r)3	r)3	X
ejpam-4854	218	10	β(7β2	β(7β2	X
ejpam-4854	219	1	+	+	NUM
ejpam-4854	219	2	9βr	9βr	ADJ
ejpam-4854	219	3	+	+	NOUN
ejpam-4854	219	4	3r2	3r2	NUM
ejpam-4854	219	5	)	)	PUNCT
ejpam-4854	220	1	6β2(2β	6β2(2β	NOUN
ejpam-4854	220	2	+	+	CCONJ
ejpam-4854	220	3	r	r	NOUN
ejpam-4854	220	4	)	)	PUNCT
ejpam-4854	220	5	6β3	6β3	NUM
ejpam-4854	221	1			NOUN
ejpam-4854	221	2			NOUN
ejpam-4854	221	3	1	1	NUM
ejpam-4854	221	4	1	1	NUM
ejpam-4854	221	5	1	1	NUM
ejpam-4854	221	6	1	1	NUM
ejpam-4854	221	7	0	0	NUM
ejpam-4854	221	8	1	1	NUM
ejpam-4854	221	9	2	2	NUM
ejpam-4854	221	10	3	3	NUM
ejpam-4854	221	11	0	0	NUM
ejpam-4854	221	12	0	0	NUM
ejpam-4854	221	13	1	1	NUM
ejpam-4854	221	14	3	3	NUM
ejpam-4854	221	15	0	0	NUM
ejpam-4854	221	16	0	0	NUM
ejpam-4854	221	17	0	0	NUM
ejpam-4854	221	18	1	1	NUM
ejpam-4854	221	19			NOUN
ejpam-4854	221	20	=	=	PUNCT
ejpam-4854	221	21			NOUN
ejpam-4854	221	22	1	1	NUM
ejpam-4854	221	23	1	1	NUM
ejpam-4854	221	24	1	1	NUM
ejpam-4854	221	25	1	1	NUM
ejpam-4854	221	26	β	β	X
ejpam-4854	221	27	+	+	NUM
ejpam-4854	221	28	r	r	NOUN
ejpam-4854	221	29	2β	2β	NOUN
ejpam-4854	221	30	+	+	CCONJ
ejpam-4854	221	31	r	r	NOUN
ejpam-4854	221	32	3β	3β	NUM
ejpam-4854	221	33	+	+	CCONJ
ejpam-4854	221	34	r	r	NOUN
ejpam-4854	221	35	4β	4β	NOUN
ejpam-4854	221	36	+	+	CCONJ
ejpam-4854	221	37	r	r	NOUN
ejpam-4854	221	38	(	(	PUNCT
ejpam-4854	221	39	β	β	NOUN
ejpam-4854	221	40	+	+	ADJ
ejpam-4854	221	41	r)2	r)2	NOUN
ejpam-4854	221	42	(	(	PUNCT
ejpam-4854	221	43	2β	2β	NOUN
ejpam-4854	221	44	+	+	CCONJ
ejpam-4854	221	45	r)2	r)2	NOUN
ejpam-4854	221	46	(	(	PUNCT
ejpam-4854	221	47	3β	3β	NUM
ejpam-4854	221	48	+	+	CCONJ
ejpam-4854	221	49	r)2	r)2	NOUN
ejpam-4854	221	50	(	(	PUNCT
ejpam-4854	221	51	4β	4β	NOUN
ejpam-4854	221	52	+	+	CCONJ
ejpam-4854	221	53	r)2	r)2	ADJ
ejpam-4854	221	54	(	(	PUNCT
ejpam-4854	221	55	β	β	X
ejpam-4854	221	56	+	+	CCONJ
ejpam-4854	221	57	r)3	r)3	NOUN
ejpam-4854	221	58	(	(	PUNCT
ejpam-4854	221	59	2β	2β	NOUN
ejpam-4854	221	60	+	+	CCONJ
ejpam-4854	221	61	r)3	r)3	NOUN
ejpam-4854	221	62	(	(	PUNCT
ejpam-4854	221	63	3β	3β	NUM
ejpam-4854	221	64	+	+	CCONJ
ejpam-4854	221	65	r)3	r)3	PROPN
ejpam-4854	221	66	(	(	PUNCT
ejpam-4854	221	67	4β	4β	NOUN
ejpam-4854	221	68	+	+	CCONJ
ejpam-4854	221	69	r)3	r)3	NOUN
ejpam-4854	221	70			NOUN
ejpam-4854	221	71	=	=	SYM
ejpam-4854	221	72	vβ	vβ	NOUN
ejpam-4854	221	73	,	,	PUNCT
ejpam-4854	221	74	r	r	NOUN
ejpam-4854	221	75	4	4	NUM
ejpam-4854	221	76	(	(	PUNCT
ejpam-4854	221	77	t	t	NOUN
ejpam-4854	221	78	)	)	PUNCT
ejpam-4854	221	79	.	.	PUNCT
ejpam-4854	222	1	corollary	corollary	ADJ
ejpam-4854	222	2	2	2	NUM
ejpam-4854	222	3	.	.	PUNCT
ejpam-4854	223	1	for	for	ADP
ejpam-4854	223	2	any	any	DET
ejpam-4854	223	3	real	real	ADJ
ejpam-4854	223	4	number	number	NOUN
ejpam-4854	223	5	t	t	PROPN
ejpam-4854	223	6	,	,	PUNCT
ejpam-4854	223	7	det(vβ	det(vβ	NOUN
ejpam-4854	223	8	,	,	PUNCT
ejpam-4854	223	9	r	r	NOUN
ejpam-4854	223	10	n	n	PROPN
ejpam-4854	223	11	(	(	PUNCT
ejpam-4854	223	12	t	t	PROPN
ejpam-4854	223	13	)	)	PUNCT
ejpam-4854	223	14	)	)	PUNCT
ejpam-4854	224	1	=	=	SYM
ejpam-4854	224	2	n−1∏	n−1∏	PROPN
ejpam-4854	224	3	k=0	k=0	PROPN
ejpam-4854	224	4	k!βk	k!βk	PROPN
ejpam-4854	224	5	.	.	PROPN
ejpam-4854	224	6	proof	proof	NOUN
ejpam-4854	224	7	.	.	PUNCT
ejpam-4854	225	1	let	let	VERB
ejpam-4854	225	2	vβ	vβ	ADP
ejpam-4854	225	3	n(t	n(t	NOUN
ejpam-4854	225	4	)	)	PUNCT
ejpam-4854	225	5	be	be	VERB
ejpam-4854	225	6	the	the	DET
ejpam-4854	225	7	generalized	generalized	ADJ
ejpam-4854	225	8	vandermonde	vandermonde	NOUN
ejpam-4854	225	9	matrix	matrix	NOUN
ejpam-4854	225	10	defined	define	VERB
ejpam-4854	225	11	by	by	ADP
ejpam-4854	225	12	,	,	PUNCT
ejpam-4854	225	13	vβ	vβ	NOUN
ejpam-4854	225	14	,	,	PUNCT
ejpam-4854	225	15	r	r	NOUN
ejpam-4854	225	16	n	n	PROPN
ejpam-4854	225	17	(	(	PUNCT
ejpam-4854	225	18	t	t	NOUN
ejpam-4854	225	19	)	)	PUNCT
ejpam-4854	225	20	=	=	SYM
ejpam-4854	226	1	vβ	vβ	NOUN
ejpam-4854	226	2	,	,	PUNCT
ejpam-4854	226	3	r	r	NOUN
ejpam-4854	226	4	n	n	NUM
ejpam-4854	226	5	(	(	PUNCT
ejpam-4854	226	6	βt+	βt+	ADJ
ejpam-4854	226	7	r	r	NOUN
ejpam-4854	226	8	,	,	PUNCT
ejpam-4854	226	9	βt+	βt+	ADJ
ejpam-4854	226	10	β	β	X
ejpam-4854	226	11	+	+	CCONJ
ejpam-4854	226	12	r	r	NOUN
ejpam-4854	226	13	,	,	PUNCT
ejpam-4854	226	14	βt+	βt+	ADJ
ejpam-4854	226	15	2β	2β	NOUN
ejpam-4854	226	16	+	+	CCONJ
ejpam-4854	226	17	r	r	NOUN
ejpam-4854	226	18	,	,	PUNCT
ejpam-4854	226	19	.	.	PUNCT
ejpam-4854	226	20	.	.	PUNCT
ejpam-4854	226	21	.	.	PUNCT
ejpam-4854	227	1	,	,	PUNCT
ejpam-4854	227	2	βt+	βt+	X
ejpam-4854	227	3	(	(	PUNCT
ejpam-4854	227	4	n−	n−	NOUN
ejpam-4854	227	5	1)β	1)β	NUM
ejpam-4854	227	6	+	+	CCONJ
ejpam-4854	227	7	r	r	NOUN
ejpam-4854	227	8	)	)	PUNCT
ejpam-4854	227	9	.	.	PUNCT
ejpam-4854	228	1	by	by	ADP
ejpam-4854	228	2	kalman	kalman	PROPN
ejpam-4854	228	3	[	[	X
ejpam-4854	228	4	11	11	NUM
ejpam-4854	228	5	]	]	PUNCT
ejpam-4854	228	6	,	,	PUNCT
ejpam-4854	228	7	the	the	DET
ejpam-4854	228	8	formula	formula	NOUN
ejpam-4854	228	9	for	for	ADP
ejpam-4854	228	10	getting	get	VERB
ejpam-4854	228	11	the	the	DET
ejpam-4854	228	12	determinant	determinant	NOUN
ejpam-4854	228	13	of	of	ADP
ejpam-4854	228	14	a	a	DET
ejpam-4854	228	15	vandermonde	vandermonde	ADJ
ejpam-4854	228	16	matris	matris	NOUN
ejpam-4854	228	17	is	be	AUX
ejpam-4854	228	18	det((v	det((v	PROPN
ejpam-4854	228	19	β	β	X
ejpam-4854	228	20	n	n	X
ejpam-4854	228	21	(	(	PUNCT
ejpam-4854	228	22	t	t	PROPN
ejpam-4854	228	23	)	)	PUNCT
ejpam-4854	228	24	)	)	PUNCT
ejpam-4854	229	1	=	=	SYM
ejpam-4854	229	2	n−1∏	n−1∏	PROPN
ejpam-4854	229	3	k=0	k=0	PROPN
ejpam-4854	229	4	(	(	PUNCT
ejpam-4854	229	5	ti	ti	NOUN
ejpam-4854	229	6	−	−	PROPN
ejpam-4854	229	7	tj	tj	NOUN
ejpam-4854	229	8	)	)	PUNCT
ejpam-4854	229	9	.	.	PUNCT
ejpam-4854	230	1	g.	g.	PROPN
ejpam-4854	230	2	engalan	engalan	PROPN
ejpam-4854	230	3	,	,	PUNCT
ejpam-4854	230	4	m.r	m.r	PROPN
ejpam-4854	230	5	.	.	PROPN
ejpam-4854	230	6	latayada	latayada	PROPN
ejpam-4854	230	7	/	/	SYM
ejpam-4854	230	8	eur	eur	PROPN
ejpam-4854	230	9	.	.	PUNCT
ejpam-4854	231	1	j.	j.	PROPN
ejpam-4854	231	2	pure	pure	PROPN
ejpam-4854	231	3	appl	appl	PROPN
ejpam-4854	231	4	.	.	PROPN
ejpam-4854	231	5	math	math	PROPN
ejpam-4854	231	6	,	,	PUNCT
ejpam-4854	231	7	16	16	NUM
ejpam-4854	231	8	(	(	PUNCT
ejpam-4854	231	9	4	4	NUM
ejpam-4854	231	10	)	)	PUNCT
ejpam-4854	231	11	(	(	PUNCT
ejpam-4854	231	12	2023	2023	NUM
ejpam-4854	231	13	)	)	PUNCT
ejpam-4854	231	14	,	,	PUNCT
ejpam-4854	231	15	2306	2306	NUM
ejpam-4854	231	16	-	-	SYM
ejpam-4854	231	17	2322	2322	NUM
ejpam-4854	231	18	2317	2317	NUM
ejpam-4854	231	19	now	now	ADV
ejpam-4854	231	20	,	,	PUNCT
ejpam-4854	231	21	det((v	det((v	PROPN
ejpam-4854	231	22	β	β	X
ejpam-4854	231	23	n	n	PROPN
ejpam-4854	231	24	(	(	PUNCT
ejpam-4854	231	25	t	t	PROPN
ejpam-4854	231	26	)	)	PUNCT
ejpam-4854	231	27	)	)	PUNCT
ejpam-4854	232	1	=	=	PUNCT
ejpam-4854	233	1	[	[	X
ejpam-4854	233	2	(	(	PUNCT
ejpam-4854	233	3	βt+	βt+	ADJ
ejpam-4854	233	4	(	(	PUNCT
ejpam-4854	233	5	n−	n−	NOUN
ejpam-4854	233	6	1)β	1)β	NUM
ejpam-4854	233	7	+	+	CCONJ
ejpam-4854	233	8	r)−	r)−	PROPN
ejpam-4854	233	9	(	(	PUNCT
ejpam-4854	233	10	βt+	βt+	ADJ
ejpam-4854	233	11	r	r	NOUN
ejpam-4854	233	12	)	)	PUNCT
ejpam-4854	233	13	]	]	PUNCT
ejpam-4854	233	14	·	·	PUNCT
ejpam-4854	233	15	·	·	PUNCT
ejpam-4854	233	16	·	·	PUNCT
ejpam-4854	234	1	[	[	X
ejpam-4854	234	2	(	(	PUNCT
ejpam-4854	234	3	βt+	βt+	ADJ
ejpam-4854	234	4	(	(	PUNCT
ejpam-4854	234	5	n−	n−	NOUN
ejpam-4854	234	6	1)β	1)β	NUM
ejpam-4854	234	7	+	+	CCONJ
ejpam-4854	234	8	r)−	r)−	PROPN
ejpam-4854	234	9	(	(	PUNCT
ejpam-4854	234	10	βt+	βt+	ADJ
ejpam-4854	234	11	(	(	PUNCT
ejpam-4854	234	12	n−	n−	NOUN
ejpam-4854	234	13	2)β	2)β	NOUN
ejpam-4854	234	14	+	+	CCONJ
ejpam-4854	234	15	r	r	X
ejpam-4854	234	16	)	)	PUNCT
ejpam-4854	234	17	]	]	PUNCT
ejpam-4854	235	1	×	×	NOUN
ejpam-4854	235	2	[	[	X
ejpam-4854	235	3	(	(	PUNCT
ejpam-4854	235	4	βt+	βt+	ADJ
ejpam-4854	235	5	(	(	PUNCT
ejpam-4854	235	6	n−	n−	NOUN
ejpam-4854	235	7	2)β	2)β	NOUN
ejpam-4854	235	8	+	+	CCONJ
ejpam-4854	236	1	r)−	r)−	PROPN
ejpam-4854	236	2	(	(	PUNCT
ejpam-4854	236	3	βt+	βt+	ADJ
ejpam-4854	236	4	r	r	NOUN
ejpam-4854	236	5	)	)	PUNCT
ejpam-4854	236	6	]	]	PUNCT
ejpam-4854	236	7	·	·	PUNCT
ejpam-4854	236	8	·	·	PUNCT
ejpam-4854	236	9	·	·	PUNCT
ejpam-4854	237	1	[	[	X
ejpam-4854	237	2	(	(	PUNCT
ejpam-4854	237	3	βt+	βt+	ADJ
ejpam-4854	237	4	(	(	PUNCT
ejpam-4854	237	5	n−	n−	NOUN
ejpam-4854	237	6	2)β	2)β	NOUN
ejpam-4854	237	7	+	+	CCONJ
ejpam-4854	238	1	r)−	r)−	PROPN
ejpam-4854	238	2	(	(	PUNCT
ejpam-4854	238	3	βt+	βt+	ADJ
ejpam-4854	238	4	(	(	PUNCT
ejpam-4854	238	5	n−	n−	NOUN
ejpam-4854	238	6	3)β	3)β	NUM
ejpam-4854	238	7	+	+	CCONJ
ejpam-4854	238	8	r	r	X
ejpam-4854	238	9	)	)	PUNCT
ejpam-4854	238	10	]	]	PUNCT
ejpam-4854	238	11	×	×	NOUN
ejpam-4854	239	1	[	[	X
ejpam-4854	239	2	(	(	PUNCT
ejpam-4854	239	3	βt+	βt+	ADJ
ejpam-4854	239	4	(	(	PUNCT
ejpam-4854	239	5	n−	n−	NOUN
ejpam-4854	239	6	3)β	3)β	NUM
ejpam-4854	239	7	+	+	CCONJ
ejpam-4854	240	1	r)−	r)−	PROPN
ejpam-4854	240	2	(	(	PUNCT
ejpam-4854	240	3	βt+	βt+	ADJ
ejpam-4854	240	4	r	r	NOUN
ejpam-4854	240	5	)	)	PUNCT
ejpam-4854	240	6	]	]	PUNCT
ejpam-4854	240	7	·	·	PUNCT
ejpam-4854	240	8	·	·	PUNCT
ejpam-4854	240	9	·	·	PUNCT
ejpam-4854	241	1	[	[	X
ejpam-4854	241	2	(	(	PUNCT
ejpam-4854	241	3	βt+	βt+	ADJ
ejpam-4854	241	4	(	(	PUNCT
ejpam-4854	241	5	n−	n−	NOUN
ejpam-4854	241	6	3)β	3)β	NUM
ejpam-4854	241	7	+	+	CCONJ
ejpam-4854	242	1	r)−	r)−	PROPN
ejpam-4854	242	2	(	(	PUNCT
ejpam-4854	242	3	βt+	βt+	ADJ
ejpam-4854	242	4	(	(	PUNCT
ejpam-4854	242	5	n−	n−	NOUN
ejpam-4854	242	6	4)β	4)β	NOUN
ejpam-4854	242	7	+	+	CCONJ
ejpam-4854	242	8	r	r	NOUN
ejpam-4854	242	9	)	)	PUNCT
ejpam-4854	242	10	]	]	PUNCT
ejpam-4854	242	11	...	...	PUNCT
ejpam-4854	243	1	×	×	PRON
ejpam-4854	244	1	[	[	X
ejpam-4854	244	2	(	(	PUNCT
ejpam-4854	244	3	βt+	βt+	ADJ
ejpam-4854	244	4	β	β	X
ejpam-4854	244	5	+	+	X
ejpam-4854	244	6	r)−	r)−	PROPN
ejpam-4854	244	7	(	(	PUNCT
ejpam-4854	244	8	βt+	βt+	ADJ
ejpam-4854	244	9	r	r	X
ejpam-4854	244	10	)	)	PUNCT
ejpam-4854	244	11	]	]	PUNCT
ejpam-4854	245	1	=	=	PUNCT
ejpam-4854	245	2	(	(	PUNCT
ejpam-4854	245	3	n−	n−	PROPN
ejpam-4854	245	4	1)β	1)β	NUM
ejpam-4854	245	5	·	·	PUNCT
ejpam-4854	245	6	(	(	PUNCT
ejpam-4854	245	7	n−	n−	NOUN
ejpam-4854	245	8	2)β	2)β	NOUN
ejpam-4854	245	9	·	·	PUNCT
ejpam-4854	245	10	(	(	PUNCT
ejpam-4854	245	11	n−	n−	NOUN
ejpam-4854	245	12	3)β	3)β	NUM
ejpam-4854	245	13	·	·	PUNCT
ejpam-4854	245	14	·	·	PUNCT
ejpam-4854	245	15	·	·	PUNCT
ejpam-4854	245	16	β	β	X
ejpam-4854	245	17	×	×	NOUN
ejpam-4854	245	18	(	(	PUNCT
ejpam-4854	245	19	n−	n−	NOUN
ejpam-4854	245	20	2)β	2)β	NOUN
ejpam-4854	245	21	·	·	PUNCT
ejpam-4854	245	22	(	(	PUNCT
ejpam-4854	245	23	n−	n−	NOUN
ejpam-4854	245	24	3)β	3)β	NUM
ejpam-4854	245	25	·	·	PUNCT
ejpam-4854	245	26	(	(	PUNCT
ejpam-4854	245	27	n−	n−	NOUN
ejpam-4854	245	28	4)β	4)β	NOUN
ejpam-4854	245	29	·	·	PUNCT
ejpam-4854	245	30	·	·	PUNCT
ejpam-4854	245	31	·	·	PUNCT
ejpam-4854	245	32	β	β	X
ejpam-4854	245	33	×	×	NOUN
ejpam-4854	245	34	(	(	PUNCT
ejpam-4854	245	35	n−	n−	NOUN
ejpam-4854	245	36	3)β	3)β	NUM
ejpam-4854	245	37	·	·	PUNCT
ejpam-4854	245	38	(	(	PUNCT
ejpam-4854	245	39	n−	n−	NOUN
ejpam-4854	245	40	4)β	4)β	NOUN
ejpam-4854	245	41	·	·	PUNCT
ejpam-4854	245	42	(	(	PUNCT
ejpam-4854	245	43	n−	n−	NOUN
ejpam-4854	245	44	5)β	5)β	NOUN
ejpam-4854	245	45	·	·	PUNCT
ejpam-4854	245	46	·	·	PUNCT
ejpam-4854	245	47	·	·	PUNCT
ejpam-4854	245	48	β	β	X
ejpam-4854	245	49	...	...	PUNCT
ejpam-4854	245	50	×	×	NOUN
ejpam-4854	245	51	β	β	X
ejpam-4854	245	52	=	=	SYM
ejpam-4854	245	53	n−1∏	n−1∏	PROPN
ejpam-4854	245	54	k=0	k=0	PROPN
ejpam-4854	245	55	k!βk	k!βk	PROPN
ejpam-4854	245	56	.	.	PROPN
ejpam-4854	245	57	example	example	NOUN
ejpam-4854	245	58	9	9	NUM
ejpam-4854	245	59	.	.	PUNCT
ejpam-4854	246	1	let	let	VERB
ejpam-4854	246	2	n	n	NOUN
ejpam-4854	246	3	=	=	SYM
ejpam-4854	246	4	4	4	X
ejpam-4854	246	5	.	.	PUNCT
ejpam-4854	247	1	then	then	ADV
ejpam-4854	247	2	we	we	PRON
ejpam-4854	247	3	have	have	VERB
ejpam-4854	247	4	,	,	PUNCT
ejpam-4854	247	5	vβ	vβ	X
ejpam-4854	247	6	,	,	PUNCT
ejpam-4854	247	7	r	r	NOUN
ejpam-4854	247	8	4	4	NUM
ejpam-4854	247	9	(	(	PUNCT
ejpam-4854	247	10	t	t	NOUN
ejpam-4854	247	11	)	)	PUNCT
ejpam-4854	247	12	=	=	NOUN
ejpam-4854	247	13			NOUN
ejpam-4854	247	14	1	1	NUM
ejpam-4854	247	15	1	1	NUM
ejpam-4854	247	16	1	1	NUM
ejpam-4854	247	17	1	1	NUM
ejpam-4854	247	18	β	β	X
ejpam-4854	247	19	+	+	NUM
ejpam-4854	247	20	r	r	NOUN
ejpam-4854	247	21	2β	2β	NOUN
ejpam-4854	247	22	+	+	CCONJ
ejpam-4854	247	23	r	r	NOUN
ejpam-4854	247	24	3β	3β	NUM
ejpam-4854	247	25	+	+	CCONJ
ejpam-4854	248	1	r	r	NOUN
ejpam-4854	248	2	4β	4β	NOUN
ejpam-4854	248	3	+	+	CCONJ
ejpam-4854	248	4	r	r	NOUN
ejpam-4854	248	5	(	(	PUNCT
ejpam-4854	248	6	β	β	NOUN
ejpam-4854	248	7	+	+	ADJ
ejpam-4854	248	8	r)2	r)2	NOUN
ejpam-4854	248	9	(	(	PUNCT
ejpam-4854	248	10	2β	2β	NOUN
ejpam-4854	248	11	+	+	CCONJ
ejpam-4854	248	12	r)2	r)2	NOUN
ejpam-4854	248	13	(	(	PUNCT
ejpam-4854	248	14	3β	3β	NUM
ejpam-4854	248	15	+	+	CCONJ
ejpam-4854	248	16	r)2	r)2	NOUN
ejpam-4854	248	17	(	(	PUNCT
ejpam-4854	248	18	4β	4β	NOUN
ejpam-4854	248	19	+	+	CCONJ
ejpam-4854	248	20	r)2	r)2	ADJ
ejpam-4854	248	21	(	(	PUNCT
ejpam-4854	248	22	β	β	X
ejpam-4854	248	23	+	+	CCONJ
ejpam-4854	248	24	r)3	r)3	NOUN
ejpam-4854	248	25	(	(	PUNCT
ejpam-4854	248	26	2β	2β	NOUN
ejpam-4854	248	27	+	+	CCONJ
ejpam-4854	248	28	r)3	r)3	NOUN
ejpam-4854	248	29	(	(	PUNCT
ejpam-4854	248	30	3β	3β	NUM
ejpam-4854	248	31	+	+	CCONJ
ejpam-4854	248	32	r)3	r)3	PROPN
ejpam-4854	248	33	(	(	PUNCT
ejpam-4854	248	34	4β	4β	NOUN
ejpam-4854	248	35	+	+	CCONJ
ejpam-4854	248	36	r)3	r)3	NOUN
ejpam-4854	248	37			NOUN
ejpam-4854	248	38	now	now	ADV
ejpam-4854	248	39	,	,	PUNCT
ejpam-4854	248	40	det(vβ	det(vβ	NOUN
ejpam-4854	248	41	,	,	PUNCT
ejpam-4854	248	42	r	r	NOUN
ejpam-4854	248	43	4	4	NUM
ejpam-4854	248	44	(	(	PUNCT
ejpam-4854	248	45	t	t	NOUN
ejpam-4854	248	46	)	)	PUNCT
ejpam-4854	248	47	)	)	PUNCT
ejpam-4854	249	1	=	=	PUNCT
ejpam-4854	249	2	(	(	PUNCT
ejpam-4854	249	3	βt+	βt+	ADJ
ejpam-4854	249	4	3β	3β	NOUN
ejpam-4854	249	5	+	+	CCONJ
ejpam-4854	249	6	r	r	NOUN
ejpam-4854	249	7	−	−	PROPN
ejpam-4854	249	8	(	(	PUNCT
ejpam-4854	249	9	βt+	βt+	ADJ
ejpam-4854	249	10	r))(βt+	r))(βt+	NOUN
ejpam-4854	249	11	3β	3β	NOUN
ejpam-4854	250	1	+	+	CCONJ
ejpam-4854	250	2	r	r	NOUN
ejpam-4854	250	3	−	−	PROPN
ejpam-4854	250	4	(	(	PUNCT
ejpam-4854	250	5	βt+	βt+	ADJ
ejpam-4854	250	6	β	β	X
ejpam-4854	250	7	+	+	CCONJ
ejpam-4854	250	8	r))(βt+	r))(βt+	NUM
ejpam-4854	251	1	3β	3β	NOUN
ejpam-4854	252	1	+	+	CCONJ
ejpam-4854	252	2	r	r	NOUN
ejpam-4854	252	3	−	−	PROPN
ejpam-4854	252	4	(	(	PUNCT
ejpam-4854	252	5	βt+	βt+	ADJ
ejpam-4854	252	6	2β	2β	NOUN
ejpam-4854	252	7	+	+	CCONJ
ejpam-4854	252	8	r	r	NOUN
ejpam-4854	252	9	)	)	PUNCT
ejpam-4854	252	10	)	)	PUNCT
ejpam-4854	252	11	×	×	NOUN
ejpam-4854	252	12	(	(	PUNCT
ejpam-4854	252	13	βt+	βt+	ADJ
ejpam-4854	252	14	2β	2β	NOUN
ejpam-4854	253	1	+	+	CCONJ
ejpam-4854	253	2	r	r	NOUN
ejpam-4854	253	3	−	−	PROPN
ejpam-4854	253	4	(	(	PUNCT
ejpam-4854	253	5	βt+	βt+	ADJ
ejpam-4854	253	6	r))(βt+	r))(βt+	NOUN
ejpam-4854	253	7	2β	2β	NOUN
ejpam-4854	253	8	+	+	CCONJ
ejpam-4854	253	9	r	r	NOUN
ejpam-4854	253	10	−	−	PROPN
ejpam-4854	253	11	(	(	PUNCT
ejpam-4854	254	1	βt+	βt+	ADJ
ejpam-4854	254	2	β	β	X
ejpam-4854	254	3	+	+	CCONJ
ejpam-4854	254	4	r	r	NOUN
ejpam-4854	254	5	)	)	PUNCT
ejpam-4854	254	6	)	)	PUNCT
ejpam-4854	254	7	×	×	NOUN
ejpam-4854	254	8	(	(	PUNCT
ejpam-4854	254	9	βt+	βt+	ADJ
ejpam-4854	255	1	β	β	X
ejpam-4854	255	2	+	+	NOUN
ejpam-4854	255	3	r	r	NOUN
ejpam-4854	255	4	−	−	PROPN
ejpam-4854	255	5	(	(	PUNCT
ejpam-4854	255	6	βt+	βt+	ADJ
ejpam-4854	255	7	r	r	NOUN
ejpam-4854	255	8	)	)	PUNCT
ejpam-4854	255	9	)	)	PUNCT
ejpam-4854	256	1	=	=	SYM
ejpam-4854	256	2	(	(	PUNCT
ejpam-4854	256	3	3β	3β	NUM
ejpam-4854	256	4	·	·	PUNCT
ejpam-4854	257	1	2β	2β	NUM
ejpam-4854	257	2	·	·	PUNCT
ejpam-4854	257	3	β)(2β	β)(2β	NOUN
ejpam-4854	257	4	·	·	PUNCT
ejpam-4854	257	5	β)(β	β)(β	NOUN
ejpam-4854	257	6	)	)	PUNCT
ejpam-4854	257	7	=	=	SYM
ejpam-4854	257	8	(	(	PUNCT
ejpam-4854	257	9	3	3	NUM
ejpam-4854	257	10	·	·	SYM
ejpam-4854	257	11	2	2	NUM
ejpam-4854	257	12	·	·	SYM
ejpam-4854	257	13	1β3)(2	1β3)(2	NUM
ejpam-4854	257	14	·	·	SYM
ejpam-4854	257	15	1β2)(β	1β2)(β	NUM
ejpam-4854	257	16	)	)	PUNCT
ejpam-4854	257	17	=	=	SYM
ejpam-4854	257	18	(	(	PUNCT
ejpam-4854	257	19	3!β3)(2!β2)(1!β	3!β3)(2!β2)(1!β	NUM
ejpam-4854	257	20	)	)	PUNCT
ejpam-4854	257	21	=	=	SYM
ejpam-4854	258	1	3∏	3∏	NUM
ejpam-4854	258	2	k=0	k=0	PROPN
ejpam-4854	258	3	k!βk	k!βk	PROPN
ejpam-4854	258	4	.	.	PUNCT
ejpam-4854	258	5	lemma	lemma	PROPN
ejpam-4854	258	6	1	1	X
ejpam-4854	258	7	.	.	PUNCT
ejpam-4854	259	1	let	let	VERB
ejpam-4854	259	2	ln[β	ln[β	PROPN
ejpam-4854	259	3	]	]	PUNCT
ejpam-4854	259	4	be	be	AUX
ejpam-4854	259	5	an	an	DET
ejpam-4854	259	6	n×	n×	PRON
ejpam-4854	259	7	n	n	NOUN
ejpam-4854	259	8	matrix	matrix	NOUN
ejpam-4854	259	9	defined	define	VERB
ejpam-4854	259	10	by	by	ADP
ejpam-4854	259	11	[	[	X
ejpam-4854	259	12	ln[β]]1≤i	ln[β]]1≤i	PROPN
ejpam-4854	259	13	,	,	PUNCT
ejpam-4854	259	14	j≤n	j≤n	X
ejpam-4854	259	15	=	=	PUNCT
ejpam-4854	259	16	(	(	PUNCT
ejpam-4854	259	17	j	j	PROPN
ejpam-4854	259	18	i−	i−	PROPN
ejpam-4854	259	19	1	1	NUM
ejpam-4854	259	20	)	)	PUNCT
ejpam-4854	259	21	βi−1(i−	βi−1(i−	NOUN
ejpam-4854	259	22	1	1	NUM
ejpam-4854	259	23	)	)	PUNCT
ejpam-4854	259	24	!	!	PUNCT
ejpam-4854	259	25	.	.	PUNCT
ejpam-4854	260	1	for	for	ADP
ejpam-4854	260	2	the	the	DET
ejpam-4854	260	3	n×	n×	PROPN
ejpam-4854	260	4	n	n	PROPN
ejpam-4854	260	5	(	(	PUNCT
ejpam-4854	260	6	r	r	NOUN
ejpam-4854	260	7	,	,	PUNCT
ejpam-4854	260	8	β)-stirling	β)-stirle	VERB
ejpam-4854	260	9	matrix	matrix	NOUN
ejpam-4854	260	10	s(β	s(β	PROPN
ejpam-4854	260	11	,	,	PUNCT
ejpam-4854	260	12	r)(n	r)(n	PROPN
ejpam-4854	260	13	)	)	PUNCT
ejpam-4854	260	14	,	,	PUNCT
ejpam-4854	260	15	vβ	vβ	X
ejpam-4854	260	16	,	,	PUNCT
ejpam-4854	260	17	r	r	NOUN
ejpam-4854	260	18	n	n	NUM
ejpam-4854	260	19	(	(	PUNCT
ejpam-4854	260	20	1	1	NUM
ejpam-4854	260	21	)	)	PUNCT
ejpam-4854	260	22	=	=	SYM
ejpam-4854	260	23	s(β	s(β	PROPN
ejpam-4854	260	24	,	,	PUNCT
ejpam-4854	260	25	r)(n)ln[β	r)(n)ln[β	PROPN
ejpam-4854	260	26	]	]	PUNCT
ejpam-4854	260	27	t	t	PROPN
ejpam-4854	260	28	.	.	PUNCT
ejpam-4854	261	1	g.	g.	PROPN
ejpam-4854	261	2	engalan	engalan	PROPN
ejpam-4854	261	3	,	,	PUNCT
ejpam-4854	261	4	m.r	m.r	PROPN
ejpam-4854	261	5	.	.	PROPN
ejpam-4854	261	6	latayada	latayada	PROPN
ejpam-4854	261	7	/	/	SYM
ejpam-4854	261	8	eur	eur	PROPN
ejpam-4854	261	9	.	.	PUNCT
ejpam-4854	262	1	j.	j.	PROPN
ejpam-4854	262	2	pure	pure	PROPN
ejpam-4854	262	3	appl	appl	PROPN
ejpam-4854	262	4	.	.	PROPN
ejpam-4854	262	5	math	math	PROPN
ejpam-4854	262	6	,	,	PUNCT
ejpam-4854	262	7	16	16	NUM
ejpam-4854	262	8	(	(	PUNCT
ejpam-4854	262	9	4	4	NUM
ejpam-4854	262	10	)	)	PUNCT
ejpam-4854	262	11	(	(	PUNCT
ejpam-4854	262	12	2023	2023	NUM
ejpam-4854	262	13	)	)	PUNCT
ejpam-4854	262	14	,	,	PUNCT
ejpam-4854	262	15	2306	2306	NUM
ejpam-4854	262	16	-	-	SYM
ejpam-4854	262	17	2322	2322	NUM
ejpam-4854	262	18	2318	2318	NUM
ejpam-4854	262	19	proof	proof	NOUN
ejpam-4854	262	20	.	.	PUNCT
ejpam-4854	263	1	let	let	VERB
ejpam-4854	263	2	ln[β	ln[β	PROPN
ejpam-4854	263	3	]	]	PUNCT
ejpam-4854	263	4	be	be	AUX
ejpam-4854	263	5	an	an	DET
ejpam-4854	263	6	n×	n×	PRON
ejpam-4854	263	7	n	n	NOUN
ejpam-4854	263	8	matrix	matrix	NOUN
ejpam-4854	263	9	defined	define	VERB
ejpam-4854	263	10	by	by	ADP
ejpam-4854	263	11	[	[	X
ejpam-4854	263	12	ln[β]]1≤i	ln[β]]1≤i	PROPN
ejpam-4854	263	13	,	,	PUNCT
ejpam-4854	263	14	j≤n	j≤n	X
ejpam-4854	263	15	=	=	PUNCT
ejpam-4854	263	16	(	(	PUNCT
ejpam-4854	263	17	j	j	PROPN
ejpam-4854	263	18	i−	i−	PROPN
ejpam-4854	263	19	1	1	NUM
ejpam-4854	263	20	)	)	PUNCT
ejpam-4854	263	21	βi−1(i−	βi−1(i−	NOUN
ejpam-4854	263	22	1	1	NUM
ejpam-4854	263	23	)	)	PUNCT
ejpam-4854	263	24	!	!	PUNCT
ejpam-4854	263	25	.	.	PUNCT
ejpam-4854	264	1	then	then	ADV
ejpam-4854	264	2	,	,	PUNCT
ejpam-4854	264	3	[	[	X
ejpam-4854	264	4	ln[β	ln[β	X
ejpam-4854	264	5	]	]	PUNCT
ejpam-4854	264	6	t	t	X
ejpam-4854	264	7	]	]	PUNCT
ejpam-4854	264	8	1≤i	1≤i	NUM
ejpam-4854	264	9	,	,	PUNCT
ejpam-4854	264	10	j≤n	j≤n	X
ejpam-4854	264	11	=	=	PUNCT
ejpam-4854	264	12	(	(	PUNCT
ejpam-4854	264	13	i−	i−	PROPN
ejpam-4854	264	14	1	1	NUM
ejpam-4854	264	15	j	j	PROPN
ejpam-4854	264	16	)	)	PUNCT
ejpam-4854	264	17	βjj	βjj	PROPN
ejpam-4854	264	18	!	!	PUNCT
ejpam-4854	264	19	.	.	PUNCT
ejpam-4854	265	1	now	now	ADV
ejpam-4854	265	2	,	,	PUNCT
ejpam-4854	265	3	[	[	X
ejpam-4854	265	4	s(β	s(β	ADJ
ejpam-4854	265	5	,	,	PUNCT
ejpam-4854	265	6	r)(n)ln[β	r)(n)ln[β	PROPN
ejpam-4854	265	7	]	]	PUNCT
ejpam-4854	265	8	t	t	X
ejpam-4854	265	9	]	]	X
ejpam-4854	265	10	ij	ij	NOUN
ejpam-4854	265	11	=	=	PUNCT
ejpam-4854	265	12	i−1∑	i−1∑	NUM
ejpam-4854	265	13	k=0	k=0	PROPN
ejpam-4854	265	14	〈	〈	PROPN
ejpam-4854	265	15	i−	i−	PROPN
ejpam-4854	265	16	1	1	NUM
ejpam-4854	265	17	k	k	PROPN
ejpam-4854	265	18	〉	〉	X
ejpam-4854	265	19	βkjk	βkjk	NOUN
ejpam-4854	266	1	(	(	PUNCT
ejpam-4854	266	2	i−	i−	PROPN
ejpam-4854	266	3	1	1	NUM
ejpam-4854	266	4	k	k	PROPN
ejpam-4854	266	5	)	)	PUNCT
ejpam-4854	266	6	.	.	PUNCT
ejpam-4854	267	1	by	by	ADP
ejpam-4854	267	2	equation	equation	NOUN
ejpam-4854	267	3	(	(	PUNCT
ejpam-4854	267	4	5	5	NUM
ejpam-4854	267	5	)	)	PUNCT
ejpam-4854	267	6	,	,	PUNCT
ejpam-4854	267	7	[	[	X
ejpam-4854	267	8	s(β	s(β	PROPN
ejpam-4854	267	9	,	,	PUNCT
ejpam-4854	267	10	r)(n)ln[β	r)(n)ln[β	PROPN
ejpam-4854	267	11	]	]	PUNCT
ejpam-4854	267	12	t	t	X
ejpam-4854	267	13	]	]	X
ejpam-4854	267	14	ij	ij	NOUN
ejpam-4854	267	15	=	=	SYM
ejpam-4854	267	16	(	(	PUNCT
ejpam-4854	267	17	βj	βj	X
ejpam-4854	268	1	+	+	CCONJ
ejpam-4854	268	2	r)i−1	r)i−1	X
ejpam-4854	268	3	=	=	SYM
ejpam-4854	268	4	(	(	PUNCT
ejpam-4854	268	5	β	β	X
ejpam-4854	268	6	+	+	X
ejpam-4854	268	7	(	(	PUNCT
ejpam-4854	268	8	j	j	PROPN
ejpam-4854	268	9	−	−	PROPN
ejpam-4854	268	10	1)β	1)β	NUM
ejpam-4854	268	11	+	+	CCONJ
ejpam-4854	268	12	r)i−1	r)i−1	NOUN
ejpam-4854	268	13	=	=	PUNCT
ejpam-4854	269	1	[	[	X
ejpam-4854	269	2	vβ	vβ	X
ejpam-4854	269	3	,	,	PUNCT
ejpam-4854	269	4	r	r	NOUN
ejpam-4854	269	5	n	n	NUM
ejpam-4854	269	6	(	(	PUNCT
ejpam-4854	269	7	1)]ij	1)]ij	NUM
ejpam-4854	269	8	.	.	PUNCT
ejpam-4854	270	1	theorem	theorem	NOUN
ejpam-4854	270	2	2	2	NUM
ejpam-4854	270	3	.	.	X
ejpam-4854	271	1	for	for	ADP
ejpam-4854	271	2	any	any	DET
ejpam-4854	271	3	real	real	ADJ
ejpam-4854	271	4	number	number	NOUN
ejpam-4854	271	5	t	t	PROPN
ejpam-4854	271	6	,	,	PUNCT
ejpam-4854	271	7	and	and	CCONJ
ejpam-4854	271	8	the	the	DET
ejpam-4854	271	9	generalized	generalized	ADJ
ejpam-4854	271	10	pascal	pascal	ADJ
ejpam-4854	271	11	matrix	matrix	NOUN
ejpam-4854	271	12	,	,	PUNCT
ejpam-4854	271	13	vβ	vβ	NOUN
ejpam-4854	271	14	,	,	PUNCT
ejpam-4854	271	15	r	r	NOUN
ejpam-4854	271	16	n	n	PROPN
ejpam-4854	271	17	(	(	PUNCT
ejpam-4854	271	18	t	t	PROPN
ejpam-4854	271	19	)	)	PUNCT
ejpam-4854	271	20	=	=	SYM
ejpam-4854	271	21	pn[β(t−	pn[β(t−	PROPN
ejpam-4854	271	22	1)]s(β	1)]s(β	NOUN
ejpam-4854	271	23	,	,	PUNCT
ejpam-4854	271	24	r)(n)ln[β	r)(n)ln[β	PROPN
ejpam-4854	271	25	]	]	PUNCT
ejpam-4854	271	26	t	t	NOUN
ejpam-4854	271	27	.	.	PUNCT
ejpam-4854	272	1	proof	proof	NOUN
ejpam-4854	272	2	.	.	PUNCT
ejpam-4854	273	1	from	from	ADP
ejpam-4854	273	2	lemma	lemma	PROPN
ejpam-4854	273	3	1	1	NUM
ejpam-4854	273	4	,	,	PUNCT
ejpam-4854	273	5	pn[β(t−	pn[β(t−	PROPN
ejpam-4854	273	6	1)]s(β	1)]s(β	NOUN
ejpam-4854	273	7	,	,	PUNCT
ejpam-4854	273	8	r)(n)ln[β	r)(n)ln[β	PROPN
ejpam-4854	273	9	]	]	PUNCT
ejpam-4854	273	10	t	t	NOUN
ejpam-4854	273	11	=	=	SYM
ejpam-4854	273	12	pn[β(t−	pn[β(t−	PROPN
ejpam-4854	273	13	1)]vβ	1)]vβ	NUM
ejpam-4854	273	14	,	,	PUNCT
ejpam-4854	273	15	r	r	NOUN
ejpam-4854	273	16	n	n	CCONJ
ejpam-4854	273	17	(	(	PUNCT
ejpam-4854	273	18	1	1	NUM
ejpam-4854	273	19	)	)	PUNCT
ejpam-4854	273	20	=	=	NOUN
ejpam-4854	273	21	i−1∑	i−1∑	NUM
ejpam-4854	273	22	k=0	k=0	PROPN
ejpam-4854	273	23	(	(	PUNCT
ejpam-4854	273	24	i−	i−	PROPN
ejpam-4854	273	25	1	1	NUM
ejpam-4854	273	26	k	k	NOUN
ejpam-4854	273	27	)	)	PUNCT
ejpam-4854	274	1	(	(	PUNCT
ejpam-4854	274	2	β(t−	β(t−	NOUN
ejpam-4854	274	3	1))i−1−k(βj	1))i−1−k(βj	X
ejpam-4854	275	1	+	+	CCONJ
ejpam-4854	275	2	r)k	r)k	X
ejpam-4854	275	3	=	=	SYM
ejpam-4854	275	4	(	(	PUNCT
ejpam-4854	275	5	β(t−	β(t−	NOUN
ejpam-4854	275	6	1	1	NUM
ejpam-4854	275	7	)	)	PUNCT
ejpam-4854	275	8	+	+	NUM
ejpam-4854	275	9	βj	βj	X
ejpam-4854	275	10	+	+	CCONJ
ejpam-4854	275	11	r)i−1	r)i−1	X
ejpam-4854	275	12	=	=	SYM
ejpam-4854	275	13	(	(	PUNCT
ejpam-4854	275	14	βt+	βt+	ADJ
ejpam-4854	275	15	β(j	β(j	NOUN
ejpam-4854	275	16	−	−	PROPN
ejpam-4854	275	17	1	1	NUM
ejpam-4854	275	18	)	)	PUNCT
ejpam-4854	275	19	+	+	CCONJ
ejpam-4854	275	20	r)i−1	r)i−1	NOUN
ejpam-4854	275	21	=	=	PUNCT
ejpam-4854	276	1	[	[	X
ejpam-4854	276	2	vβ	vβ	X
ejpam-4854	276	3	,	,	PUNCT
ejpam-4854	276	4	r	r	NOUN
ejpam-4854	276	5	n	n	NUM
ejpam-4854	276	6	(	(	PUNCT
ejpam-4854	276	7	x)]ij	x)]ij	PROPN
ejpam-4854	276	8	.	.	PUNCT
ejpam-4854	276	9	example	example	NOUN
ejpam-4854	277	1	10	10	NUM
ejpam-4854	277	2	.	.	PUNCT
ejpam-4854	278	1	let	let	VERB
ejpam-4854	278	2	n	n	NOUN
ejpam-4854	278	3	=	=	SYM
ejpam-4854	278	4	4	4	X
ejpam-4854	278	5	.	.	PUNCT
ejpam-4854	279	1	then	then	ADV
ejpam-4854	279	2	,	,	PUNCT
ejpam-4854	279	3	p4[β(t−	p4[β(t−	PROPN
ejpam-4854	279	4	1	1	NUM
ejpam-4854	279	5	)	)	PUNCT
ejpam-4854	279	6	]	]	PUNCT
ejpam-4854	280	1	=	=	SYM
ejpam-4854	280	2			NOUN
ejpam-4854	280	3	1	1	NUM
ejpam-4854	280	4	0	0	NUM
ejpam-4854	280	5	0	0	NUM
ejpam-4854	280	6	0	0	NUM
ejpam-4854	280	7	β(t−	β(t−	NOUN
ejpam-4854	280	8	1	1	NUM
ejpam-4854	280	9	)	)	PUNCT
ejpam-4854	280	10	1	1	NUM
ejpam-4854	280	11	0	0	NUM
ejpam-4854	280	12	0	0	NUM
ejpam-4854	281	1	(	(	PUNCT
ejpam-4854	281	2	β(t−	β(t−	NOUN
ejpam-4854	281	3	1))2	1))2	NUM
ejpam-4854	281	4	2β(t−	2β(t−	NUM
ejpam-4854	281	5	1	1	NUM
ejpam-4854	281	6	)	)	PUNCT
ejpam-4854	281	7	1	1	NUM
ejpam-4854	281	8	0	0	NUM
ejpam-4854	282	1	(	(	PUNCT
ejpam-4854	282	2	β(t−	β(t−	NOUN
ejpam-4854	282	3	1))3	1))3	NUM
ejpam-4854	283	1	3(β(t−	3(β(t−	NUM
ejpam-4854	283	2	1))2	1))2	NUM
ejpam-4854	283	3	3β(t−	3β(t−	NUM
ejpam-4854	283	4	1	1	NUM
ejpam-4854	283	5	)	)	PUNCT
ejpam-4854	283	6	1	1	NUM
ejpam-4854	283	7			NOUN
ejpam-4854	283	8	.	.	PUNCT
ejpam-4854	284	1	g.	g.	PROPN
ejpam-4854	284	2	engalan	engalan	PROPN
ejpam-4854	284	3	,	,	PUNCT
ejpam-4854	284	4	m.r	m.r	PROPN
ejpam-4854	284	5	.	.	PROPN
ejpam-4854	284	6	latayada	latayada	PROPN
ejpam-4854	284	7	/	/	SYM
ejpam-4854	284	8	eur	eur	PROPN
ejpam-4854	284	9	.	.	PUNCT
ejpam-4854	285	1	j.	j.	PROPN
ejpam-4854	285	2	pure	pure	PROPN
ejpam-4854	285	3	appl	appl	PROPN
ejpam-4854	285	4	.	.	PROPN
ejpam-4854	285	5	math	math	PROPN
ejpam-4854	285	6	,	,	PUNCT
ejpam-4854	285	7	16	16	NUM
ejpam-4854	285	8	(	(	PUNCT
ejpam-4854	285	9	4	4	NUM
ejpam-4854	285	10	)	)	PUNCT
ejpam-4854	285	11	(	(	PUNCT
ejpam-4854	285	12	2023	2023	NUM
ejpam-4854	285	13	)	)	PUNCT
ejpam-4854	285	14	,	,	PUNCT
ejpam-4854	285	15	2306	2306	NUM
ejpam-4854	285	16	-	-	SYM
ejpam-4854	285	17	2322	2322	NUM
ejpam-4854	285	18	2319	2319	NUM
ejpam-4854	285	19	s(β	s(β	PROPN
ejpam-4854	285	20	,	,	PUNCT
ejpam-4854	285	21	r)(4	r)(4	NUM
ejpam-4854	285	22	)	)	PUNCT
ejpam-4854	286	1	=	=	SYM
ejpam-4854	286	2			NOUN
ejpam-4854	287	1	1	1	NUM
ejpam-4854	287	2	0	0	NUM
ejpam-4854	287	3	0	0	NUM
ejpam-4854	287	4	0	0	NUM
ejpam-4854	288	1	r	r	NOUN
ejpam-4854	288	2	1	1	NUM
ejpam-4854	288	3	0	0	NUM
ejpam-4854	288	4	0	0	NUM
ejpam-4854	288	5	r2	r2	NOUN
ejpam-4854	288	6	β	β	X
ejpam-4854	288	7	+	+	CCONJ
ejpam-4854	288	8	2r	2r	NUM
ejpam-4854	288	9	1	1	NUM
ejpam-4854	288	10	0	0	NUM
ejpam-4854	288	11	r3	r3	NOUN
ejpam-4854	288	12	β3	β3	NOUN
ejpam-4854	288	13	+	+	CCONJ
ejpam-4854	288	14	3βr	3βr	ADJ
ejpam-4854	288	15	+	+	CCONJ
ejpam-4854	288	16	r2	r2	PROPN
ejpam-4854	288	17	3β	3β	NOUN
ejpam-4854	288	18	+	+	CCONJ
ejpam-4854	288	19	3r	3r	NUM
ejpam-4854	288	20	1	1	NUM
ejpam-4854	288	21			NOUN
ejpam-4854	288	22	,	,	PUNCT
ejpam-4854	288	23	and	and	CCONJ
ejpam-4854	288	24	l4[β	l4[β	PROPN
ejpam-4854	288	25	]	]	X
ejpam-4854	288	26	t	t	NOUN
ejpam-4854	288	27	=	=	SYM
ejpam-4854	288	28			NOUN
ejpam-4854	288	29	1	1	NUM
ejpam-4854	288	30	1	1	NUM
ejpam-4854	288	31	1	1	NUM
ejpam-4854	288	32	1	1	NUM
ejpam-4854	288	33	β	β	NOUN
ejpam-4854	288	34	2β	2β	NOUN
ejpam-4854	288	35	3β	3β	NUM
ejpam-4854	288	36	4β	4β	NOUN
ejpam-4854	288	37	0	0	NUM
ejpam-4854	288	38	2β2	2β2	NUM
ejpam-4854	288	39	6β2	6β2	NUM
ejpam-4854	288	40	12β2	12β2	NUM
ejpam-4854	288	41	0	0	NUM
ejpam-4854	288	42	0	0	NUM
ejpam-4854	288	43	6β3	6β3	NUM
ejpam-4854	288	44	24β3	24β3	NUM
ejpam-4854	288	45			NOUN
ejpam-4854	288	46	.	.	PUNCT
ejpam-4854	289	1	now	now	ADV
ejpam-4854	289	2	,	,	PUNCT
ejpam-4854	289	3	p4[β(t−	p4[β(t−	PROPN
ejpam-4854	289	4	1)]s(β	1)]s(β	PROPN
ejpam-4854	289	5	,	,	PUNCT
ejpam-4854	289	6	r)(4)l4[β	r)(4)l4[β	X
ejpam-4854	289	7	]	]	X
ejpam-4854	289	8	t	t	NOUN
ejpam-4854	289	9	=	=	SYM
ejpam-4854	289	10			NOUN
ejpam-4854	289	11	1	1	NUM
ejpam-4854	289	12	0	0	NUM
ejpam-4854	289	13	0	0	NUM
ejpam-4854	289	14	0	0	NUM
ejpam-4854	290	1	β(t−	β(t−	NOUN
ejpam-4854	290	2	1	1	NUM
ejpam-4854	290	3	)	)	PUNCT
ejpam-4854	290	4	1	1	NUM
ejpam-4854	290	5	0	0	NUM
ejpam-4854	290	6	0	0	NUM
ejpam-4854	291	1	(	(	PUNCT
ejpam-4854	291	2	β(t−	β(t−	NOUN
ejpam-4854	291	3	1))2	1))2	NUM
ejpam-4854	291	4	2β(t−	2β(t−	NUM
ejpam-4854	291	5	1	1	NUM
ejpam-4854	291	6	)	)	PUNCT
ejpam-4854	291	7	1	1	NUM
ejpam-4854	291	8	0	0	NUM
ejpam-4854	292	1	(	(	PUNCT
ejpam-4854	292	2	β(t−	β(t−	NOUN
ejpam-4854	292	3	1))3	1))3	NUM
ejpam-4854	293	1	3(β(t−	3(β(t−	NUM
ejpam-4854	293	2	1))2	1))2	NUM
ejpam-4854	293	3	3β(t−	3β(t−	NUM
ejpam-4854	293	4	1	1	NUM
ejpam-4854	293	5	)	)	PUNCT
ejpam-4854	293	6	1	1	NUM
ejpam-4854	293	7			NOUN
ejpam-4854	293	8	×	×	NOUN
ejpam-4854	293	9			NOUN
ejpam-4854	293	10	1	1	NUM
ejpam-4854	293	11	0	0	NUM
ejpam-4854	293	12	0	0	NUM
ejpam-4854	293	13	0	0	NUM
ejpam-4854	293	14	r	r	NOUN
ejpam-4854	293	15	1	1	NUM
ejpam-4854	293	16	0	0	NUM
ejpam-4854	293	17	0	0	NUM
ejpam-4854	293	18	r2	r2	NOUN
ejpam-4854	293	19	β	β	X
ejpam-4854	293	20	+	+	CCONJ
ejpam-4854	293	21	2r	2r	NUM
ejpam-4854	293	22	1	1	NUM
ejpam-4854	293	23	0	0	NUM
ejpam-4854	293	24	r3	r3	NOUN
ejpam-4854	293	25	β3	β3	NOUN
ejpam-4854	293	26	+	+	CCONJ
ejpam-4854	293	27	3βr	3βr	ADJ
ejpam-4854	293	28	+	+	CCONJ
ejpam-4854	293	29	r2	r2	PROPN
ejpam-4854	293	30	3β	3β	NOUN
ejpam-4854	293	31	+	+	CCONJ
ejpam-4854	293	32	3r	3r	NUM
ejpam-4854	293	33	1	1	NUM
ejpam-4854	293	34			NOUN
ejpam-4854	293	35			NOUN
ejpam-4854	293	36	1	1	NUM
ejpam-4854	293	37	1	1	NUM
ejpam-4854	293	38	1	1	NUM
ejpam-4854	293	39	1	1	NUM
ejpam-4854	293	40	β	β	NOUN
ejpam-4854	293	41	2β	2β	NOUN
ejpam-4854	293	42	3β	3β	NUM
ejpam-4854	293	43	4β	4β	NOUN
ejpam-4854	293	44	0	0	NUM
ejpam-4854	293	45	2β2	2β2	NUM
ejpam-4854	293	46	6β2	6β2	NUM
ejpam-4854	293	47	12β2	12β2	NUM
ejpam-4854	293	48	0	0	NUM
ejpam-4854	293	49	0	0	NUM
ejpam-4854	293	50	6β3	6β3	NUM
ejpam-4854	293	51	24β3	24β3	NUM
ejpam-4854	293	52			NOUN
ejpam-4854	293	53	=	=	SYM
ejpam-4854	293	54			NOUN
ejpam-4854	294	1	1	1	NUM
ejpam-4854	294	2	0	0	NUM
ejpam-4854	294	3	0	0	NUM
ejpam-4854	294	4	0	0	NUM
ejpam-4854	294	5	β(t−	β(t−	NOUN
ejpam-4854	294	6	1	1	NUM
ejpam-4854	294	7	)	)	PUNCT
ejpam-4854	294	8	1	1	NUM
ejpam-4854	294	9	0	0	NUM
ejpam-4854	294	10	0	0	NUM
ejpam-4854	295	1	(	(	PUNCT
ejpam-4854	295	2	β(t−	β(t−	NOUN
ejpam-4854	295	3	1))2	1))2	NUM
ejpam-4854	295	4	2β(t−	2β(t−	NUM
ejpam-4854	295	5	1	1	NUM
ejpam-4854	295	6	)	)	PUNCT
ejpam-4854	295	7	1	1	NUM
ejpam-4854	295	8	0	0	NUM
ejpam-4854	296	1	(	(	PUNCT
ejpam-4854	296	2	β(t−	β(t−	NOUN
ejpam-4854	296	3	1))3	1))3	NUM
ejpam-4854	297	1	3(β(t−	3(β(t−	NUM
ejpam-4854	297	2	1))2	1))2	NUM
ejpam-4854	297	3	3β(t−	3β(t−	NUM
ejpam-4854	297	4	1	1	NUM
ejpam-4854	297	5	)	)	PUNCT
ejpam-4854	297	6	1	1	NUM
ejpam-4854	297	7			NOUN
ejpam-4854	297	8	×	×	NOUN
ejpam-4854	297	9			NOUN
ejpam-4854	297	10	1	1	NUM
ejpam-4854	297	11	1	1	NUM
ejpam-4854	297	12	1	1	NUM
ejpam-4854	297	13	1	1	NUM
ejpam-4854	297	14	β	β	X
ejpam-4854	297	15	+	+	NUM
ejpam-4854	297	16	r	r	NOUN
ejpam-4854	297	17	2β	2β	NOUN
ejpam-4854	297	18	+	+	CCONJ
ejpam-4854	297	19	r	r	NOUN
ejpam-4854	297	20	0	0	NUM
ejpam-4854	297	21	0	0	NUM
ejpam-4854	297	22	(	(	PUNCT
ejpam-4854	297	23	β	β	X
ejpam-4854	297	24	+	+	ADJ
ejpam-4854	297	25	r)2	r)2	NOUN
ejpam-4854	297	26	(	(	PUNCT
ejpam-4854	297	27	2β	2β	NOUN
ejpam-4854	297	28	+	+	CCONJ
ejpam-4854	297	29	r)2	r)2	NOUN
ejpam-4854	297	30	(	(	PUNCT
ejpam-4854	297	31	3β	3β	NUM
ejpam-4854	297	32	+	+	CCONJ
ejpam-4854	297	33	r)2	r)2	NOUN
ejpam-4854	297	34	(	(	PUNCT
ejpam-4854	297	35	4β	4β	NOUN
ejpam-4854	297	36	+	+	CCONJ
ejpam-4854	297	37	r)2	r)2	ADJ
ejpam-4854	297	38	(	(	PUNCT
ejpam-4854	297	39	β	β	X
ejpam-4854	297	40	+	+	CCONJ
ejpam-4854	297	41	r)3	r)3	NOUN
ejpam-4854	297	42	(	(	PUNCT
ejpam-4854	297	43	2β	2β	NOUN
ejpam-4854	297	44	+	+	CCONJ
ejpam-4854	297	45	r)3	r)3	NOUN
ejpam-4854	297	46	(	(	PUNCT
ejpam-4854	297	47	3β	3β	NUM
ejpam-4854	297	48	+	+	CCONJ
ejpam-4854	297	49	r)3	r)3	PROPN
ejpam-4854	297	50	(	(	PUNCT
ejpam-4854	297	51	4β	4β	NOUN
ejpam-4854	297	52	+	+	CCONJ
ejpam-4854	297	53	r)3	r)3	NOUN
ejpam-4854	297	54			NOUN
ejpam-4854	297	55	=	=	SYM
ejpam-4854	297	56			NOUN
ejpam-4854	297	57	1	1	NUM
ejpam-4854	297	58	1	1	NUM
ejpam-4854	297	59	1	1	NUM
ejpam-4854	297	60	1	1	NUM
ejpam-4854	297	61	βt+	βt+	NOUN
ejpam-4854	297	62	r	r	NOUN
ejpam-4854	297	63	βt+	βt+	X
ejpam-4854	298	1	β	β	X
ejpam-4854	298	2	+	+	NOUN
ejpam-4854	298	3	r	r	NOUN
ejpam-4854	298	4	βt+	βt+	ADJ
ejpam-4854	298	5	2β	2β	NOUN
ejpam-4854	299	1	+	+	CCONJ
ejpam-4854	299	2	r	r	NOUN
ejpam-4854	299	3	βt+	βt+	ADJ
ejpam-4854	299	4	3β	3β	NOUN
ejpam-4854	299	5	+	+	CCONJ
ejpam-4854	300	1	r	r	NOUN
ejpam-4854	300	2	(	(	PUNCT
ejpam-4854	300	3	βt+	βt+	ADJ
ejpam-4854	300	4	r)2	r)2	NOUN
ejpam-4854	300	5	(	(	PUNCT
ejpam-4854	300	6	βt+	βt+	ADJ
ejpam-4854	300	7	β	β	X
ejpam-4854	300	8	+	+	ADJ
ejpam-4854	300	9	r)2	r)2	NOUN
ejpam-4854	300	10	(	(	PUNCT
ejpam-4854	300	11	βt+	βt+	ADJ
ejpam-4854	300	12	2β	2β	NOUN
ejpam-4854	300	13	+	+	CCONJ
ejpam-4854	300	14	r)2	r)2	NOUN
ejpam-4854	300	15	(	(	PUNCT
ejpam-4854	300	16	βt+	βt+	ADJ
ejpam-4854	300	17	3β	3β	NOUN
ejpam-4854	300	18	+	+	CCONJ
ejpam-4854	300	19	r)2	r)2	NOUN
ejpam-4854	300	20	(	(	PUNCT
ejpam-4854	300	21	βt+	βt+	ADJ
ejpam-4854	300	22	r)3	r)3	NOUN
ejpam-4854	300	23	(	(	PUNCT
ejpam-4854	300	24	βt+	βt+	ADJ
ejpam-4854	300	25	β	β	X
ejpam-4854	300	26	+	+	CCONJ
ejpam-4854	300	27	r)3	r)3	NOUN
ejpam-4854	300	28	(	(	PUNCT
ejpam-4854	300	29	βt+	βt+	ADJ
ejpam-4854	300	30	2β	2β	NOUN
ejpam-4854	300	31	+	+	CCONJ
ejpam-4854	300	32	r)3	r)3	PROPN
ejpam-4854	300	33	(	(	PUNCT
ejpam-4854	300	34	βt+	βt+	ADJ
ejpam-4854	300	35	3β	3β	NOUN
ejpam-4854	300	36	+	+	CCONJ
ejpam-4854	300	37	r)3	r)3	NOUN
ejpam-4854	300	38			NOUN
ejpam-4854	300	39	=	=	SYM
ejpam-4854	300	40	vβ	vβ	NOUN
ejpam-4854	300	41	,	,	PUNCT
ejpam-4854	300	42	r	r	NOUN
ejpam-4854	300	43	4	4	NUM
ejpam-4854	300	44	(	(	PUNCT
ejpam-4854	300	45	t	t	PROPN
ejpam-4854	300	46	)	)	PUNCT
ejpam-4854	300	47	.	.	PUNCT
ejpam-4854	301	1	2.4	2.4	NUM
ejpam-4854	301	2	.	.	PUNCT
ejpam-4854	301	3	successive	successive	ADJ
ejpam-4854	301	4	sum	sum	NOUN
ejpam-4854	301	5	of	of	ADP
ejpam-4854	301	6	powers	power	NOUN
ejpam-4854	301	7	of	of	ADP
ejpam-4854	301	8	arithmetic	arithmetic	ADJ
ejpam-4854	301	9	progressions	progression	NOUN
ejpam-4854	301	10	we	we	PRON
ejpam-4854	301	11	will	will	AUX
ejpam-4854	301	12	show	show	VERB
ejpam-4854	301	13	that	that	DET
ejpam-4854	301	14	matrix	matrix	NOUN
ejpam-4854	301	15	s̃(β	s̃(β	PROPN
ejpam-4854	301	16	,	,	PUNCT
ejpam-4854	301	17	r)(n	r)(n	PROPN
ejpam-4854	301	18	)	)	PUNCT
ejpam-4854	301	19	can	can	AUX
ejpam-4854	301	20	be	be	AUX
ejpam-4854	301	21	used	use	VERB
ejpam-4854	301	22	to	to	PART
ejpam-4854	301	23	derive	derive	VERB
ejpam-4854	301	24	a	a	DET
ejpam-4854	301	25	summation	summation	NOUN
ejpam-4854	301	26	formula	formula	NOUN
ejpam-4854	301	27	for	for	ADP
ejpam-4854	301	28	arithmetic	arithmetic	ADJ
ejpam-4854	301	29	progressions	progression	NOUN
ejpam-4854	301	30	.	.	PUNCT
ejpam-4854	302	1	the	the	DET
ejpam-4854	302	2	following	follow	VERB
ejpam-4854	302	3	equations	equation	NOUN
ejpam-4854	302	4	defined	define	VERB
ejpam-4854	302	5	by	by	ADP
ejpam-4854	302	6	bazsó	bazsó	NOUN
ejpam-4854	302	7	and	and	CCONJ
ejpam-4854	302	8	pintér	pintér	NOUN
ejpam-4854	302	9	in	in	ADP
ejpam-4854	302	10	[	[	X
ejpam-4854	302	11	1	1	NUM
ejpam-4854	302	12	]	]	PUNCT
ejpam-4854	302	13	,	,	PUNCT
ejpam-4854	302	14	and	and	CCONJ
ejpam-4854	302	15	mezo	mezo	PROPN
ejpam-4854	302	16	and	and	CCONJ
ejpam-4854	302	17	ramı́rez	ramı́rez	PROPN
ejpam-4854	302	18	in	in	ADP
ejpam-4854	302	19	[	[	X
ejpam-4854	302	20	12	12	NUM
ejpam-4854	302	21	]	]	PUNCT
ejpam-4854	302	22	will	will	AUX
ejpam-4854	302	23	be	be	AUX
ejpam-4854	302	24	utilized	utilize	VERB
ejpam-4854	302	25	for	for	ADP
ejpam-4854	302	26	the	the	DET
ejpam-4854	302	27	proof	proof	NOUN
ejpam-4854	302	28	of	of	ADP
ejpam-4854	302	29	the	the	DET
ejpam-4854	302	30	following	follow	VERB
ejpam-4854	302	31	theorem	theorem	PROPN
ejpam-4854	302	32	.	.	PUNCT
ejpam-4854	303	1	g.	g.	PROPN
ejpam-4854	303	2	engalan	engalan	PROPN
ejpam-4854	303	3	,	,	PUNCT
ejpam-4854	303	4	m.r	m.r	PROPN
ejpam-4854	303	5	.	.	PROPN
ejpam-4854	303	6	latayada	latayada	PROPN
ejpam-4854	303	7	/	/	SYM
ejpam-4854	303	8	eur	eur	PROPN
ejpam-4854	303	9	.	.	PUNCT
ejpam-4854	304	1	j.	j.	PROPN
ejpam-4854	304	2	pure	pure	PROPN
ejpam-4854	304	3	appl	appl	PROPN
ejpam-4854	304	4	.	.	PROPN
ejpam-4854	304	5	math	math	PROPN
ejpam-4854	304	6	,	,	PUNCT
ejpam-4854	304	7	16	16	NUM
ejpam-4854	304	8	(	(	PUNCT
ejpam-4854	304	9	4	4	NUM
ejpam-4854	304	10	)	)	PUNCT
ejpam-4854	304	11	(	(	PUNCT
ejpam-4854	304	12	2023	2023	NUM
ejpam-4854	304	13	)	)	PUNCT
ejpam-4854	304	14	,	,	PUNCT
ejpam-4854	304	15	2306	2306	NUM
ejpam-4854	304	16	-	-	SYM
ejpam-4854	304	17	2322	2322	NUM
ejpam-4854	304	18	2320	2320	NUM
ejpam-4854	304	19	definition	definition	NOUN
ejpam-4854	304	20	3	3	NUM
ejpam-4854	304	21	.	.	PUNCT
ejpam-4854	305	1	[	[	X
ejpam-4854	305	2	1	1	X
ejpam-4854	305	3	]	]	PUNCT
ejpam-4854	305	4	for	for	ADP
ejpam-4854	305	5	k	k	PROPN
ejpam-4854	305	6	=	=	SYM
ejpam-4854	305	7	1	1	NUM
ejpam-4854	305	8	,	,	PUNCT
ejpam-4854	305	9	2	2	NUM
ejpam-4854	305	10	,	,	PUNCT
ejpam-4854	305	11	.	.	PUNCT
ejpam-4854	305	12	.	.	PUNCT
ejpam-4854	305	13	.	.	PUNCT
ejpam-4854	306	1	,	,	PUNCT
ejpam-4854	306	2	n	n	CCONJ
ejpam-4854	306	3	,	,	PUNCT
ejpam-4854	306	4	the	the	DET
ejpam-4854	306	5	numbers	number	NOUN
ejpam-4854	306	6	zk	zk	PROPN
ejpam-4854	306	7	1,β	1,β	NUM
ejpam-4854	306	8	,	,	PUNCT
ejpam-4854	306	9	r(l	r(l	NOUN
ejpam-4854	306	10	)	)	PUNCT
ejpam-4854	306	11	,	,	PUNCT
ejpam-4854	306	12	z	z	NOUN
ejpam-4854	306	13	k	k	X
ejpam-4854	307	1	2,β	2,β	ADV
ejpam-4854	307	2	,	,	PUNCT
ejpam-4854	307	3	r(l	r(l	NOUN
ejpam-4854	307	4	)	)	PUNCT
ejpam-4854	307	5	are	be	AUX
ejpam-4854	307	6	defined	define	VERB
ejpam-4854	307	7	by	by	ADP
ejpam-4854	307	8	the	the	DET
ejpam-4854	307	9	recursive	recursive	ADJ
ejpam-4854	307	10	formulas	formula	NOUN
ejpam-4854	307	11	:	:	PUNCT
ejpam-4854	307	12	zk	zk	PROPN
ejpam-4854	307	13	1,β	1,β	NUM
ejpam-4854	307	14	,	,	PUNCT
ejpam-4854	307	15	r(l	r(l	NOUN
ejpam-4854	307	16	)	)	PUNCT
ejpam-4854	307	17	=	=	SYM
ejpam-4854	307	18	rk	rk	NOUN
ejpam-4854	307	19	+	+	CCONJ
ejpam-4854	307	20	(	(	PUNCT
ejpam-4854	307	21	β	β	X
ejpam-4854	307	22	+	+	X
ejpam-4854	307	23	r)k	r)k	X
ejpam-4854	308	1	+	+	CCONJ
ejpam-4854	308	2	(	(	PUNCT
ejpam-4854	308	3	2β	2β	NOUN
ejpam-4854	308	4	+	+	CCONJ
ejpam-4854	308	5	r)k	r)k	X
ejpam-4854	308	6	+	+	CCONJ
ejpam-4854	308	7	·	·	PUNCT
ejpam-4854	308	8	·	·	PUNCT
ejpam-4854	308	9	·	·	PUNCT
ejpam-4854	308	10	+	+	PUNCT
ejpam-4854	308	11	(	(	PUNCT
ejpam-4854	308	12	(	(	PUNCT
ejpam-4854	308	13	l	l	NOUN
ejpam-4854	308	14	−	−	PROPN
ejpam-4854	308	15	1)β	1)β	NUM
ejpam-4854	308	16	+	+	CCONJ
ejpam-4854	308	17	r)k	r)k	PUNCT
ejpam-4854	308	18	=	=	SYM
ejpam-4854	308	19	l−1∑	l−1∑	X
ejpam-4854	308	20	j=0	j=0	PROPN
ejpam-4854	308	21	(	(	PUNCT
ejpam-4854	308	22	jβ	jβ	PROPN
ejpam-4854	308	23	+	+	CCONJ
ejpam-4854	308	24	r)k	r)k	X
ejpam-4854	308	25	,	,	PUNCT
ejpam-4854	308	26	(	(	PUNCT
ejpam-4854	308	27	8)	8)	NUM
ejpam-4854	308	28	zk	zk	PROPN
ejpam-4854	308	29	p	p	PROPN
ejpam-4854	308	30	,	,	PUNCT
ejpam-4854	308	31	β	β	X
ejpam-4854	308	32	,	,	PUNCT
ejpam-4854	308	33	r(l	r(l	NOUN
ejpam-4854	308	34	)	)	PUNCT
ejpam-4854	308	35	=	=	SYM
ejpam-4854	308	36	n∑	n∑	NOUN
ejpam-4854	308	37	j=1	j=1	PROPN
ejpam-4854	308	38	zk	zk	PROPN
ejpam-4854	308	39	p−1,β	p−1,β	PROPN
ejpam-4854	308	40	,	,	PUNCT
ejpam-4854	308	41	r(l	r(l	NOUN
ejpam-4854	308	42	)	)	PUNCT
ejpam-4854	308	43	.	.	PUNCT
ejpam-4854	309	1	(	(	PUNCT
ejpam-4854	309	2	p	p	NOUN
ejpam-4854	309	3	=	=	SYM
ejpam-4854	309	4	2	2	NUM
ejpam-4854	309	5	,	,	PUNCT
ejpam-4854	309	6	3	3	NUM
ejpam-4854	309	7	,	,	PUNCT
ejpam-4854	309	8	4	4	NUM
ejpam-4854	309	9	,	,	PUNCT
ejpam-4854	309	10	.	.	PUNCT
ejpam-4854	309	11	.	.	PUNCT
ejpam-4854	309	12	.	.	PUNCT
ejpam-4854	309	13	)	)	PUNCT
ejpam-4854	309	14	.	.	PUNCT
ejpam-4854	310	1	(	(	PUNCT
ejpam-4854	310	2	9	9	X
ejpam-4854	310	3	)	)	PUNCT
ejpam-4854	310	4	note	note	NOUN
ejpam-4854	310	5	that	that	SCONJ
ejpam-4854	310	6	if	if	SCONJ
ejpam-4854	310	7	p	p	X
ejpam-4854	310	8	=	=	X
ejpam-4854	310	9	β	β	X
ejpam-4854	310	10	=	=	PUNCT
ejpam-4854	310	11	r	r	NOUN
ejpam-4854	310	12	=	=	SYM
ejpam-4854	310	13	1	1	NUM
ejpam-4854	310	14	,	,	PUNCT
ejpam-4854	310	15	we	we	PRON
ejpam-4854	310	16	obtain	obtain	VERB
ejpam-4854	310	17	the	the	DET
ejpam-4854	310	18	sum	sum	NOUN
ejpam-4854	310	19	of	of	ADP
ejpam-4854	310	20	powers	power	NOUN
ejpam-4854	310	21	of	of	ADP
ejpam-4854	310	22	the	the	DET
ejpam-4854	310	23	first	first	ADJ
ejpam-4854	310	24	n	n	CCONJ
ejpam-4854	310	25	positive	positive	ADJ
ejpam-4854	310	26	integers	integer	NOUN
ejpam-4854	310	27	,	,	PUNCT
ejpam-4854	310	28	that	that	PRON
ejpam-4854	310	29	is	be	AUX
ejpam-4854	310	30	zk	zk	PROPN
ejpam-4854	310	31	1,1,1(l	1,1,1(l	NUM
ejpam-4854	310	32	)	)	PUNCT
ejpam-4854	310	33	=	=	PUNCT
ejpam-4854	311	1	1k	1k	NUM
ejpam-4854	311	2	+	+	CCONJ
ejpam-4854	311	3	2k	2k	NUM
ejpam-4854	311	4	+	+	CCONJ
ejpam-4854	311	5	3k	3k	PRON
ejpam-4854	312	1	+	+	CCONJ
ejpam-4854	312	2	4k	4k	X
ejpam-4854	312	3	+	+	X
ejpam-4854	312	4	·	·	PUNCT
ejpam-4854	312	5	·	·	PUNCT
ejpam-4854	312	6	·	·	PUNCT
ejpam-4854	312	7	+	+	NUM
ejpam-4854	312	8	lk	lk	ADJ
ejpam-4854	312	9	definition	definition	NOUN
ejpam-4854	312	10	4	4	NUM
ejpam-4854	312	11	.	.	PUNCT
ejpam-4854	313	1	[	[	X
ejpam-4854	313	2	12	12	NUM
ejpam-4854	313	3	]	]	PUNCT
ejpam-4854	313	4	for	for	ADP
ejpam-4854	313	5	each	each	DET
ejpam-4854	313	6	i	i	NOUN
ejpam-4854	313	7	=	=	NOUN
ejpam-4854	313	8	1	1	NUM
ejpam-4854	313	9	,	,	PUNCT
ejpam-4854	313	10	2	2	NUM
ejpam-4854	313	11	,	,	PUNCT
ejpam-4854	313	12	.	.	PUNCT
ejpam-4854	313	13	.	.	PUNCT
ejpam-4854	314	1	.	.	PUNCT
ejpam-4854	315	1	,	,	PUNCT
ejpam-4854	316	1	n	n	CCONJ
ejpam-4854	316	2	,	,	PUNCT
ejpam-4854	316	3	and	and	CCONJ
ejpam-4854	316	4	for	for	ADP
ejpam-4854	316	5	p	p	PRON
ejpam-4854	316	6	≥	≥	NOUN
ejpam-4854	316	7	0	0	NUM
ejpam-4854	316	8	,	,	PUNCT
ejpam-4854	316	9	ti(p	ti(p	NOUN
ejpam-4854	316	10	)	)	PUNCT
ejpam-4854	317	1	=	=	PUNCT
ejpam-4854	318	1	[	[	X
ejpam-4854	318	2	(	(	PUNCT
ejpam-4854	318	3	p+	p+	NOUN
ejpam-4854	318	4	i−	i−	PROPN
ejpam-4854	318	5	2	2	NUM
ejpam-4854	318	6	p−	p−	NOUN
ejpam-4854	318	7	1	1	NUM
ejpam-4854	318	8	)	)	PUNCT
ejpam-4854	318	9	,	,	PUNCT
ejpam-4854	318	10	(	(	PUNCT
ejpam-4854	318	11	p+	p+	VERB
ejpam-4854	318	12	i−	i−	PROPN
ejpam-4854	318	13	2	2	NUM
ejpam-4854	318	14	p	p	NOUN
ejpam-4854	318	15	)	)	PUNCT
ejpam-4854	318	16	,	,	PUNCT
ejpam-4854	318	17	.	.	PUNCT
ejpam-4854	318	18	.	.	PUNCT
ejpam-4854	319	1	.	.	PUNCT
ejpam-4854	320	1	,	,	PUNCT
ejpam-4854	320	2	(	(	PUNCT
ejpam-4854	320	3	p+	p+	AUX
ejpam-4854	320	4	i−	i−	PROPN
ejpam-4854	320	5	2	2	NUM
ejpam-4854	320	6	p+	p+	NOUN
ejpam-4854	320	7	k	k	NOUN
ejpam-4854	320	8	−	−	PROPN
ejpam-4854	320	9	2	2	NUM
ejpam-4854	320	10	)	)	PUNCT
ejpam-4854	320	11	]	]	X
ejpam-4854	320	12	t	t	X
ejpam-4854	320	13	(	(	PUNCT
ejpam-4854	320	14	10	10	NUM
ejpam-4854	320	15	)	)	PUNCT
ejpam-4854	320	16	zβ	zβ	PROPN
ejpam-4854	320	17	,	,	PUNCT
ejpam-4854	320	18	ri	ri	PROPN
ejpam-4854	320	19	(	(	PUNCT
ejpam-4854	320	20	p	p	NOUN
ejpam-4854	320	21	)	)	PUNCT
ejpam-4854	320	22	=	=	NOUN
ejpam-4854	321	1	[	[	X
ejpam-4854	321	2	(	(	PUNCT
ejpam-4854	321	3	p+	p+	NOUN
ejpam-4854	321	4	i−	i−	PROPN
ejpam-4854	321	5	2	2	NUM
ejpam-4854	321	6	p−	p−	NOUN
ejpam-4854	321	7	1	1	NUM
ejpam-4854	321	8	)	)	PUNCT
ejpam-4854	321	9	,	,	PUNCT
ejpam-4854	321	10	z1	z1	PROPN
ejpam-4854	321	11	p	p	PROPN
ejpam-4854	321	12	,	,	PUNCT
ejpam-4854	321	13	β	β	X
ejpam-4854	321	14	,	,	PUNCT
ejpam-4854	321	15	r(i	r(i	NOUN
ejpam-4854	321	16	)	)	PUNCT
ejpam-4854	321	17	,	,	PUNCT
ejpam-4854	321	18	.	.	PUNCT
ejpam-4854	321	19	.	.	PUNCT
ejpam-4854	322	1	.	.	PUNCT
ejpam-4854	323	1	,	,	PUNCT
ejpam-4854	323	2	z	z	NOUN
ejpam-4854	323	3	k−1	k−1	PROPN
ejpam-4854	323	4	p−1,β	p−1,β	NOUN
ejpam-4854	323	5	,	,	PUNCT
ejpam-4854	323	6	r(i	r(i	X
ejpam-4854	323	7	)	)	PUNCT
ejpam-4854	323	8	]	]	X
ejpam-4854	323	9	t	t	X
ejpam-4854	323	10	(	(	PUNCT
ejpam-4854	323	11	11	11	NUM
ejpam-4854	323	12	)	)	PUNCT
ejpam-4854	323	13	theorem	theorem	NOUN
ejpam-4854	323	14	3	3	NUM
ejpam-4854	323	15	.	.	X
ejpam-4854	323	16	for	for	ADP
ejpam-4854	323	17	each	each	DET
ejpam-4854	323	18	p	p	NOUN
ejpam-4854	323	19	=	=	SYM
ejpam-4854	323	20	1	1	NUM
ejpam-4854	323	21	,	,	PUNCT
ejpam-4854	323	22	2	2	NUM
ejpam-4854	323	23	,	,	PUNCT
ejpam-4854	323	24	3	3	NUM
ejpam-4854	323	25	,	,	PUNCT
ejpam-4854	323	26	.	.	PUNCT
ejpam-4854	323	27	.	.	PUNCT
ejpam-4854	324	1	.	.	PUNCT
ejpam-4854	325	1	,	,	PUNCT
ejpam-4854	325	2	n	n	CCONJ
ejpam-4854	325	3	,	,	PUNCT
ejpam-4854	325	4	we	we	PRON
ejpam-4854	325	5	have	have	AUX
ejpam-4854	325	6	s̃β	s̃β	VERB
ejpam-4854	325	7	,	,	PUNCT
ejpam-4854	325	8	r(k	r(k	PROPN
ejpam-4854	325	9	)	)	PUNCT
ejpam-4854	326	1	[	[	X
ejpam-4854	326	2	(	(	PUNCT
ejpam-4854	326	3	n+p−1	n+p−1	ADJ
ejpam-4854	326	4	p	p	NOUN
ejpam-4854	326	5	)	)	PUNCT
ejpam-4854	326	6	,	,	PUNCT
ejpam-4854	326	7	β	β	X
ejpam-4854	326	8	(	(	PUNCT
ejpam-4854	326	9	n+p−1	n+p−1	PROPN
ejpam-4854	326	10	p+1	p+1	NOUN
ejpam-4854	326	11	)	)	PUNCT
ejpam-4854	326	12	,	,	PUNCT
ejpam-4854	326	13	.	.	PUNCT
ejpam-4854	326	14	.	.	PUNCT
ejpam-4854	326	15	.	.	PUNCT
ejpam-4854	327	1	,	,	PUNCT
ejpam-4854	327	2	βk−1	βk−1	INTJ
ejpam-4854	327	3	(	(	PUNCT
ejpam-4854	327	4	n+p−1	n+p−1	X
ejpam-4854	327	5	p+k−1	p+k−1	PROPN
ejpam-4854	327	6	)	)	PUNCT
ejpam-4854	327	7	]	]	PUNCT
ejpam-4854	328	1	t	t	X
ejpam-4854	329	1	=	=	PUNCT
ejpam-4854	330	1	[	[	X
ejpam-4854	330	2	(	(	PUNCT
ejpam-4854	330	3	n+p−1	n+p−1	ADJ
ejpam-4854	330	4	p	p	NOUN
ejpam-4854	330	5	)	)	PUNCT
ejpam-4854	330	6	,	,	PUNCT
ejpam-4854	330	7	z1	z1	PROPN
ejpam-4854	330	8	p	p	PROPN
ejpam-4854	330	9	,	,	PUNCT
ejpam-4854	330	10	β	β	X
ejpam-4854	330	11	,	,	PUNCT
ejpam-4854	330	12	r(n	r(n	PROPN
ejpam-4854	330	13	)	)	PUNCT
ejpam-4854	330	14	,	,	PUNCT
ejpam-4854	330	15	.	.	PUNCT
ejpam-4854	330	16	.	.	PUNCT
ejpam-4854	331	1	.	.	PUNCT
ejpam-4854	332	1	,	,	PUNCT
ejpam-4854	332	2	z	z	PROPN
ejpam-4854	332	3	k−	k−	PROPN
ejpam-4854	332	4	p	p	NOUN
ejpam-4854	332	5	,	,	PUNCT
ejpam-4854	332	6	β	β	X
ejpam-4854	332	7	,	,	PUNCT
ejpam-4854	332	8	r(n	r(n	PROPN
ejpam-4854	332	9	)	)	PUNCT
ejpam-4854	332	10	]	]	X
ejpam-4854	332	11	t	t	X
ejpam-4854	332	12	(	(	PUNCT
ejpam-4854	332	13	12	12	NUM
ejpam-4854	332	14	)	)	PUNCT
ejpam-4854	332	15	proof	proof	NOUN
ejpam-4854	332	16	.	.	PUNCT
ejpam-4854	333	1	let	let	VERB
ejpam-4854	333	2	n	n	PRON
ejpam-4854	333	3	and	and	CCONJ
ejpam-4854	333	4	k	k	PROPN
ejpam-4854	333	5	be	be	AUX
ejpam-4854	333	6	positive	positive	ADJ
ejpam-4854	333	7	integers	integer	NOUN
ejpam-4854	334	1	such	such	ADJ
ejpam-4854	334	2	that	that	SCONJ
ejpam-4854	334	3	n	n	NUM
ejpam-4854	334	4	≥	≥	NOUN
ejpam-4854	334	5	k	k	NOUN
ejpam-4854	334	6	and	and	CCONJ
ejpam-4854	334	7	p	p	NOUN
ejpam-4854	334	8	=	=	PROPN
ejpam-4854	334	9	1	1	NUM
ejpam-4854	334	10	,	,	PUNCT
ejpam-4854	334	11	2	2	NUM
ejpam-4854	334	12	,	,	PUNCT
ejpam-4854	334	13	3	3	NUM
ejpam-4854	334	14	,	,	PUNCT
ejpam-4854	334	15	.	.	PUNCT
ejpam-4854	334	16	.	.	PUNCT
ejpam-4854	334	17	.	.	PUNCT
ejpam-4854	335	1	,	,	PUNCT
ejpam-4854	335	2	n.	n.	PROPN
ejpam-4854	335	3	now	now	ADV
ejpam-4854	335	4	,	,	PUNCT
ejpam-4854	335	5	we	we	PRON
ejpam-4854	335	6	will	will	AUX
ejpam-4854	335	7	prove	prove	VERB
ejpam-4854	335	8	equation	equation	NOUN
ejpam-4854	335	9	(	(	PUNCT
ejpam-4854	335	10	3.7	3.7	NUM
ejpam-4854	335	11	)	)	PUNCT
ejpam-4854	335	12	by	by	ADP
ejpam-4854	335	13	induction	induction	NOUN
ejpam-4854	335	14	on	on	ADP
ejpam-4854	335	15	n+	n+	PUNCT
ejpam-4854	335	16	p.	p.	NOUN
ejpam-4854	335	17	note	note	VERB
ejpam-4854	335	18	that	that	SCONJ
ejpam-4854	335	19	the	the	DET
ejpam-4854	335	20	sum	sum	NOUN
ejpam-4854	335	21	of	of	ADP
ejpam-4854	335	22	the	the	DET
ejpam-4854	335	23	entries	entry	NOUN
ejpam-4854	335	24	in	in	ADP
ejpam-4854	335	25	the	the	DET
ejpam-4854	335	26	second	second	ADJ
ejpam-4854	335	27	row	row	NOUN
ejpam-4854	335	28	of	of	ADP
ejpam-4854	335	29	vβ	vβ	ADJ
ejpam-4854	335	30	,	,	PUNCT
ejpam-4854	335	31	r	r	NOUN
ejpam-4854	335	32	n	n	NUM
ejpam-4854	335	33	(	(	PUNCT
ejpam-4854	335	34	0	0	NUM
ejpam-4854	335	35	)	)	PUNCT
ejpam-4854	335	36	is	be	AUX
ejpam-4854	335	37	r	r	NOUN
ejpam-4854	335	38	+	+	CCONJ
ejpam-4854	335	39	(	(	PUNCT
ejpam-4854	335	40	β	β	X
ejpam-4854	335	41	+	+	X
ejpam-4854	336	1	r	r	X
ejpam-4854	336	2	)	)	PUNCT
ejpam-4854	336	3	+	+	CCONJ
ejpam-4854	336	4	(	(	PUNCT
ejpam-4854	336	5	2β	2β	NOUN
ejpam-4854	336	6	+	+	CCONJ
ejpam-4854	336	7	r	r	NOUN
ejpam-4854	336	8	)	)	PUNCT
ejpam-4854	336	9	+	+	NUM
ejpam-4854	336	10	·	·	PUNCT
ejpam-4854	336	11	·	·	PUNCT
ejpam-4854	336	12	·	·	PUNCT
ejpam-4854	336	13	+	+	PUNCT
ejpam-4854	336	14	(	(	PUNCT
ejpam-4854	336	15	(	(	PUNCT
ejpam-4854	336	16	n−	n−	NOUN
ejpam-4854	336	17	1)β	1)β	NUM
ejpam-4854	336	18	+	+	CCONJ
ejpam-4854	336	19	r	r	NOUN
ejpam-4854	336	20	)	)	PUNCT
ejpam-4854	336	21	=	=	SYM
ejpam-4854	336	22	r1	r1	PROPN
ejpam-4854	336	23	+	+	CCONJ
ejpam-4854	336	24	(	(	PUNCT
ejpam-4854	336	25	β	β	X
ejpam-4854	336	26	+	+	CCONJ
ejpam-4854	336	27	r)1	r)1	PROPN
ejpam-4854	336	28	+	+	CCONJ
ejpam-4854	336	29	(	(	PUNCT
ejpam-4854	336	30	2β	2β	NOUN
ejpam-4854	336	31	+	+	CCONJ
ejpam-4854	336	32	r)1	r)1	PROPN
ejpam-4854	336	33	+	+	CCONJ
ejpam-4854	336	34	·	·	PUNCT
ejpam-4854	336	35	·	·	PUNCT
ejpam-4854	336	36	·	·	PUNCT
ejpam-4854	337	1	+	+	PUNCT
ejpam-4854	337	2	(	(	PUNCT
ejpam-4854	337	3	(	(	PUNCT
ejpam-4854	337	4	n−	n−	NOUN
ejpam-4854	337	5	1)β	1)β	NUM
ejpam-4854	337	6	+	+	CCONJ
ejpam-4854	337	7	r)1	r)1	PROPN
ejpam-4854	337	8	=	=	SYM
ejpam-4854	337	9	zk	zk	PROPN
ejpam-4854	337	10	1,β	1,β	NUM
ejpam-4854	337	11	,	,	PUNCT
ejpam-4854	337	12	r(n	r(n	PROPN
ejpam-4854	337	13	)	)	PUNCT
ejpam-4854	337	14	.	.	PUNCT
ejpam-4854	338	1	now	now	ADV
ejpam-4854	338	2	,	,	PUNCT
ejpam-4854	338	3	substituting	substitute	VERB
ejpam-4854	338	4	t	t	NOUN
ejpam-4854	338	5	=	=	SYM
ejpam-4854	338	6	0	0	NUM
ejpam-4854	338	7	to	to	ADP
ejpam-4854	338	8	equation	equation	NOUN
ejpam-4854	338	9	(	(	PUNCT
ejpam-4854	338	10	12	12	NUM
ejpam-4854	338	11	)	)	PUNCT
ejpam-4854	338	12	,	,	PUNCT
ejpam-4854	338	13	we	we	PRON
ejpam-4854	338	14	have	have	VERB
ejpam-4854	338	15	vβ	vβ	ADJ
ejpam-4854	338	16	,	,	PUNCT
ejpam-4854	338	17	r	r	NOUN
ejpam-4854	338	18	n	n	NUM
ejpam-4854	338	19	(	(	PUNCT
ejpam-4854	338	20	0	0	NUM
ejpam-4854	338	21	)	)	PUNCT
ejpam-4854	338	22	=	=	VERB
ejpam-4854	338	23	s̃β	s̃β	VERB
ejpam-4854	338	24	,	,	PUNCT
ejpam-4854	338	25	rcβ	rcβ	NOUN
ejpam-4854	338	26	n	n	CCONJ
ejpam-4854	338	27	(	(	PUNCT
ejpam-4854	338	28	0	0	NUM
ejpam-4854	338	29	)	)	PUNCT
ejpam-4854	338	30	=	=	PUNCT
ejpam-4854	338	31	s̃β	s̃β	VERB
ejpam-4854	338	32	,	,	PUNCT
ejpam-4854	338	33	r	r	X
ejpam-4854	339	1	[	[	X
ejpam-4854	339	2	(	(	PUNCT
ejpam-4854	339	3	n	n	NOUN
ejpam-4854	339	4	1	1	NUM
ejpam-4854	339	5	)	)	PUNCT
ejpam-4854	339	6	,	,	PUNCT
ejpam-4854	339	7	β	β	X
ejpam-4854	339	8	(	(	PUNCT
ejpam-4854	339	9	n	n	PROPN
ejpam-4854	339	10	2	2	NUM
ejpam-4854	339	11	)	)	PUNCT
ejpam-4854	339	12	,	,	PUNCT
ejpam-4854	339	13	.	.	PUNCT
ejpam-4854	339	14	.	.	PUNCT
ejpam-4854	340	1	.	.	PUNCT
ejpam-4854	341	1	,	,	PUNCT
ejpam-4854	341	2	βk−1	βk−1	INTJ
ejpam-4854	341	3	(	(	PUNCT
ejpam-4854	341	4	n−1	n−1	PROPN
ejpam-4854	341	5	k−1	k−1	PROPN
ejpam-4854	341	6	)	)	PUNCT
ejpam-4854	342	1	]	]	X
ejpam-4854	342	2	t	t	X
ejpam-4854	343	1	=	=	PUNCT
ejpam-4854	344	1	[	[	X
ejpam-4854	344	2	(	(	PUNCT
ejpam-4854	344	3	n−1	n−1	PROPN
ejpam-4854	344	4	1	1	NUM
ejpam-4854	344	5	)	)	PUNCT
ejpam-4854	344	6	,	,	PUNCT
ejpam-4854	344	7	z1	z1	NOUN
ejpam-4854	344	8	1,β	1,β	NUM
ejpam-4854	344	9	,	,	PUNCT
ejpam-4854	344	10	r(n	r(n	PROPN
ejpam-4854	344	11	)	)	PUNCT
ejpam-4854	344	12	,	,	PUNCT
ejpam-4854	344	13	.	.	PUNCT
ejpam-4854	344	14	.	.	PUNCT
ejpam-4854	345	1	.	.	PUNCT
ejpam-4854	346	1	,	,	PUNCT
ejpam-4854	346	2	z	z	PROPN
ejpam-4854	346	3	k−1	k−1	PROPN
ejpam-4854	346	4	1,β	1,β	NOUN
ejpam-4854	346	5	,	,	PUNCT
ejpam-4854	346	6	r(n	r(n	PROPN
ejpam-4854	346	7	)	)	PUNCT
ejpam-4854	346	8	]	]	PUNCT
ejpam-4854	346	9	t	t	PROPN
ejpam-4854	346	10	.	.	PUNCT
ejpam-4854	347	1	thus	thus	ADV
ejpam-4854	347	2	,	,	PUNCT
ejpam-4854	347	3	equation	equation	NOUN
ejpam-4854	347	4	(	(	PUNCT
ejpam-4854	347	5	12	12	NUM
ejpam-4854	347	6	)	)	PUNCT
ejpam-4854	347	7	is	be	AUX
ejpam-4854	347	8	true	true	ADJ
ejpam-4854	347	9	for	for	ADP
ejpam-4854	347	10	p	p	NOUN
ejpam-4854	347	11	=	=	SYM
ejpam-4854	347	12	1	1	X
ejpam-4854	347	13	.	.	X
ejpam-4854	348	1	consider	consider	VERB
ejpam-4854	348	2	p	p	NOUN
ejpam-4854	348	3	≥	≥	NOUN
ejpam-4854	348	4	2	2	NUM
ejpam-4854	348	5	,	,	PUNCT
ejpam-4854	348	6	and	and	CCONJ
ejpam-4854	348	7	supposed	suppose	VERB
ejpam-4854	348	8	the	the	DET
ejpam-4854	348	9	result	result	NOUN
ejpam-4854	348	10	is	be	AUX
ejpam-4854	348	11	true	true	ADJ
ejpam-4854	348	12	for	for	ADP
ejpam-4854	348	13	all	all	DET
ejpam-4854	348	14	i	i	PRON
ejpam-4854	348	15	≤	≤	PUNCT
ejpam-4854	348	16	n+	n+	PUNCT
ejpam-4854	349	1	p.	p.	NOUN
ejpam-4854	349	2	using	use	VERB
ejpam-4854	349	3	the	the	DET
ejpam-4854	349	4	identity	identity	NOUN
ejpam-4854	349	5	(	(	PUNCT
ejpam-4854	349	6	n+	n+	ADP
ejpam-4854	349	7	1	1	NUM
ejpam-4854	349	8	k	k	NOUN
ejpam-4854	349	9	+	+	NOUN
ejpam-4854	349	10	1	1	X
ejpam-4854	349	11	)	)	PUNCT
ejpam-4854	350	1	=	=	SYM
ejpam-4854	350	2	n∑	n∑	PROPN
ejpam-4854	350	3	l−0	l−0	PROPN
ejpam-4854	350	4	(	(	PUNCT
ejpam-4854	350	5	l	l	NOUN
ejpam-4854	350	6	k	k	PROPN
ejpam-4854	350	7	)	)	PUNCT
ejpam-4854	350	8	,	,	PUNCT
ejpam-4854	350	9	g.	g.	PROPN
ejpam-4854	350	10	engalan	engalan	PROPN
ejpam-4854	350	11	,	,	PUNCT
ejpam-4854	350	12	m.r	m.r	PROPN
ejpam-4854	350	13	.	.	PROPN
ejpam-4854	350	14	latayada	latayada	PROPN
ejpam-4854	350	15	/	/	SYM
ejpam-4854	350	16	eur	eur	PROPN
ejpam-4854	350	17	.	.	PUNCT
ejpam-4854	351	1	j.	j.	PROPN
ejpam-4854	351	2	pure	pure	PROPN
ejpam-4854	351	3	appl	appl	PROPN
ejpam-4854	351	4	.	.	PROPN
ejpam-4854	351	5	math	math	PROPN
ejpam-4854	351	6	,	,	PUNCT
ejpam-4854	351	7	16	16	NUM
ejpam-4854	351	8	(	(	PUNCT
ejpam-4854	351	9	4	4	NUM
ejpam-4854	351	10	)	)	PUNCT
ejpam-4854	351	11	(	(	PUNCT
ejpam-4854	351	12	2023	2023	NUM
ejpam-4854	351	13	)	)	PUNCT
ejpam-4854	351	14	,	,	PUNCT
ejpam-4854	351	15	2306	2306	NUM
ejpam-4854	351	16	-	-	SYM
ejpam-4854	351	17	2322	2322	NUM
ejpam-4854	351	18	2321	2321	NUM
ejpam-4854	351	19	and	and	CCONJ
ejpam-4854	351	20	equations	equation	NOUN
ejpam-4854	351	21	(	(	PUNCT
ejpam-4854	351	22	10	10	NUM
ejpam-4854	351	23	)	)	PUNCT
ejpam-4854	351	24	and	and	CCONJ
ejpam-4854	351	25	(	(	PUNCT
ejpam-4854	351	26	11	11	NUM
ejpam-4854	351	27	)	)	PUNCT
ejpam-4854	351	28	,	,	PUNCT
ejpam-4854	351	29	by	by	ADP
ejpam-4854	351	30	induction	induction	NOUN
ejpam-4854	351	31	we	we	PRON
ejpam-4854	351	32	have	have	VERB
ejpam-4854	351	33	,	,	PUNCT
ejpam-4854	351	34	s̃β	s̃β	VERB
ejpam-4854	351	35	,	,	PUNCT
ejpam-4854	351	36	r(k)ti(p+	r(k)ti(p+	PROPN
ejpam-4854	351	37	1	1	NUM
ejpam-4854	351	38	)	)	PUNCT
ejpam-4854	351	39	=	=	PUNCT
ejpam-4854	351	40	s̃β	s̃β	VERB
ejpam-4854	351	41	,	,	PUNCT
ejpam-4854	351	42	r(k)(t1(p	r(k)(t1(p	NOUN
ejpam-4854	351	43	)	)	PUNCT
ejpam-4854	352	1	+	+	CCONJ
ejpam-4854	352	2	t2(p	t2(p	X
ejpam-4854	352	3	)	)	PUNCT
ejpam-4854	352	4	+	+	NUM
ejpam-4854	352	5	·	·	PUNCT
ejpam-4854	352	6	·	·	PUNCT
ejpam-4854	352	7	·	·	PUNCT
ejpam-4854	352	8	+	+	NUM
ejpam-4854	352	9	tn(p	tn(p	NOUN
ejpam-4854	352	10	)	)	PUNCT
ejpam-4854	352	11	)	)	PUNCT
ejpam-4854	353	1	=	=	SYM
ejpam-4854	353	2	zβ	zβ	PROPN
ejpam-4854	353	3	,	,	PUNCT
ejpam-4854	353	4	r	r	NOUN
ejpam-4854	353	5	1	1	NUM
ejpam-4854	353	6	(	(	PUNCT
ejpam-4854	353	7	p	p	NOUN
ejpam-4854	353	8	)	)	PUNCT
ejpam-4854	353	9	+	+	CCONJ
ejpam-4854	353	10	zβ	zβ	PROPN
ejpam-4854	353	11	,	,	PUNCT
ejpam-4854	353	12	r	r	NOUN
ejpam-4854	353	13	2	2	NUM
ejpam-4854	353	14	(	(	PUNCT
ejpam-4854	353	15	p	p	NOUN
ejpam-4854	353	16	)	)	PUNCT
ejpam-4854	353	17	+	+	CCONJ
ejpam-4854	353	18	·	·	PUNCT
ejpam-4854	353	19	·	·	PUNCT
ejpam-4854	353	20	·	·	PUNCT
ejpam-4854	354	1	+	+	NUM
ejpam-4854	354	2	zβ	zβ	PROPN
ejpam-4854	354	3	,	,	PUNCT
ejpam-4854	355	1	r	r	NOUN
ejpam-4854	355	2	n	n	PROPN
ejpam-4854	355	3	(	(	PUNCT
ejpam-4854	355	4	p	p	X
ejpam-4854	355	5	)	)	PUNCT
ejpam-4854	355	6	=	=	SYM
ejpam-4854	355	7	zβ	zβ	PROPN
ejpam-4854	355	8	,	,	PUNCT
ejpam-4854	355	9	r	r	NOUN
ejpam-4854	355	10	n	n	PROPN
ejpam-4854	355	11	(	(	PUNCT
ejpam-4854	355	12	p+	p+	NOUN
ejpam-4854	355	13	1	1	NUM
ejpam-4854	355	14	)	)	PUNCT
ejpam-4854	355	15	.	.	PUNCT
ejpam-4854	356	1	thus	thus	ADV
ejpam-4854	356	2	,	,	PUNCT
ejpam-4854	356	3	equation	equation	NOUN
ejpam-4854	356	4	(	(	PUNCT
ejpam-4854	356	5	12	12	NUM
ejpam-4854	356	6	)	)	PUNCT
ejpam-4854	356	7	follows	follow	VERB
ejpam-4854	356	8	.	.	PUNCT
ejpam-4854	357	1	example	example	NOUN
ejpam-4854	357	2	11	11	NUM
ejpam-4854	357	3	.	.	PUNCT
ejpam-4854	358	1	equation	equation	NOUN
ejpam-4854	358	2	(	(	PUNCT
ejpam-4854	358	3	12	12	NUM
ejpam-4854	358	4	)	)	PUNCT
ejpam-4854	358	5	in	in	ADP
ejpam-4854	358	6	theorem	theorem	ADJ
ejpam-4854	358	7	3	3	NUM
ejpam-4854	358	8	yields	yield	NOUN
ejpam-4854	358	9	nice	nice	ADJ
ejpam-4854	358	10	formulas	formula	NOUN
ejpam-4854	358	11	to	to	ADP
ejpam-4854	358	12	sums	sum	NOUN
ejpam-4854	358	13	of	of	ADP
ejpam-4854	358	14	powers	power	NOUN
ejpam-4854	358	15	of	of	ADP
ejpam-4854	358	16	integers	integer	NOUN
ejpam-4854	358	17	.	.	PUNCT
ejpam-4854	359	1	for	for	ADP
ejpam-4854	359	2	example	example	NOUN
ejpam-4854	359	3	,	,	PUNCT
ejpam-4854	359	4	if	if	SCONJ
ejpam-4854	359	5	p	p	NOUN
ejpam-4854	359	6	=	=	NOUN
ejpam-4854	359	7	1	1	NUM
ejpam-4854	359	8	,	,	PUNCT
ejpam-4854	359	9	and	and	CCONJ
ejpam-4854	359	10	k	k	PROPN
ejpam-4854	359	11	=	=	SYM
ejpam-4854	359	12	3	3	NUM
ejpam-4854	359	13	,	,	PUNCT
ejpam-4854	360	1	we	we	PRON
ejpam-4854	360	2	obtain	obtain	VERB
ejpam-4854	360	3	1	1	NUM
ejpam-4854	361	1	0	0	NUM
ejpam-4854	361	2	0	0	NUM
ejpam-4854	361	3	0	0	NUM
ejpam-4854	361	4	r	r	NOUN
ejpam-4854	361	5	1	1	NUM
ejpam-4854	361	6	0	0	NUM
ejpam-4854	361	7	0	0	NUM
ejpam-4854	361	8	r2	r2	NOUN
ejpam-4854	361	9	β	β	X
ejpam-4854	361	10	+	+	CCONJ
ejpam-4854	361	11	2r	2r	NUM
ejpam-4854	361	12	2	2	NUM
ejpam-4854	361	13	0	0	NUM
ejpam-4854	361	14	r3	r3	NOUN
ejpam-4854	361	15	β3	β3	NOUN
ejpam-4854	362	1	+	+	X
ejpam-4854	362	2	3βr	3βr	ADJ
ejpam-4854	362	3	+	+	CCONJ
ejpam-4854	362	4	3r2	3r2	NUM
ejpam-4854	362	5	6β	6β	NOUN
ejpam-4854	362	6	+	+	CCONJ
ejpam-4854	362	7	6r	6r	NUM
ejpam-4854	362	8	6	6	NUM
ejpam-4854	362	9			NOUN
ejpam-4854	362	10			NOUN
ejpam-4854	362	11	(	(	PUNCT
ejpam-4854	362	12	n	n	NOUN
ejpam-4854	362	13	1	1	NUM
ejpam-4854	362	14	)	)	PUNCT
ejpam-4854	362	15	β	β	NOUN
ejpam-4854	362	16	(	(	PUNCT
ejpam-4854	362	17	n	n	CCONJ
ejpam-4854	362	18	2	2	X
ejpam-4854	362	19	)	)	PUNCT
ejpam-4854	362	20	β2	β2	NOUN
ejpam-4854	362	21	(	(	PUNCT
ejpam-4854	362	22	n	n	CCONJ
ejpam-4854	362	23	3	3	NUM
ejpam-4854	362	24	)	)	PUNCT
ejpam-4854	362	25	β3	β3	PROPN
ejpam-4854	362	26	(	(	PUNCT
ejpam-4854	362	27	n	n	ADV
ejpam-4854	362	28	4	4	NUM
ejpam-4854	362	29	)	)	PUNCT
ejpam-4854	362	30			NOUN
ejpam-4854	362	31	=	=	SYM
ejpam-4854	362	32			NOUN
ejpam-4854	362	33	n	n	CCONJ
ejpam-4854	362	34	1	1	NUM
ejpam-4854	362	35	2n(βn−	2n(βn−	PROPN
ejpam-4854	362	36	β	β	X
ejpam-4854	363	1	+	+	CCONJ
ejpam-4854	363	2	2r	2r	NUM
ejpam-4854	363	3	1	1	NUM
ejpam-4854	363	4	6n(2β	6n(2β	NUM
ejpam-4854	363	5	2n2	2n2	NUM
ejpam-4854	363	6	−	−	NUM
ejpam-4854	363	7	3β2n+	3β2n+	NUM
ejpam-4854	363	8	β2	β2	NOUN
ejpam-4854	363	9	+	+	CCONJ
ejpam-4854	363	10	6βnr	6βnr	NUM
ejpam-4854	363	11	−	−	PROPN
ejpam-4854	363	12	6βr	6βr	NOUN
ejpam-4854	363	13	+	+	CCONJ
ejpam-4854	363	14	6r2	6r2	NUM
ejpam-4854	363	15	)	)	PUNCT
ejpam-4854	363	16	1	1	NUM
ejpam-4854	363	17	4n(βn−	4n(βn−	PROPN
ejpam-4854	363	18	β	β	X
ejpam-4854	363	19	+	+	NOUN
ejpam-4854	363	20	2r)(β2n2	2r)(β2n2	NOUN
ejpam-4854	363	21	−	−	NOUN
ejpam-4854	363	22	β2n+	β2n+	NOUN
ejpam-4854	364	1	2βnr	2βnr	NUM
ejpam-4854	365	1	−	−	NOUN
ejpam-4854	365	2	2βr	2βr	ADJ
ejpam-4854	366	1	+	+	CCONJ
ejpam-4854	366	2	2r2	2r2	X
ejpam-4854	366	3	)	)	PUNCT
ejpam-4854	366	4			NOUN
ejpam-4854	366	5	=	=	PUNCT
ejpam-4854	367	1	[	[	X
ejpam-4854	367	2	n	n	CCONJ
ejpam-4854	367	3	,	,	PUNCT
ejpam-4854	367	4	z1	z1	NOUN
ejpam-4854	367	5	1,β	1,β	NUM
ejpam-4854	367	6	,	,	PUNCT
ejpam-4854	367	7	r(n	r(n	PROPN
ejpam-4854	367	8	)	)	PUNCT
ejpam-4854	367	9	,	,	PUNCT
ejpam-4854	367	10	z	z	NOUN
ejpam-4854	367	11	2	2	NUM
ejpam-4854	367	12	1,β	1,β	NUM
ejpam-4854	367	13	,	,	PUNCT
ejpam-4854	367	14	r(n	r(n	PROPN
ejpam-4854	367	15	)	)	PUNCT
ejpam-4854	367	16	,	,	PUNCT
ejpam-4854	367	17	z	z	NOUN
ejpam-4854	367	18	3	3	NUM
ejpam-4854	367	19	1,β	1,β	NUM
ejpam-4854	367	20	,	,	PUNCT
ejpam-4854	367	21	r(n	r(n	PROPN
ejpam-4854	367	22	)	)	PUNCT
ejpam-4854	367	23	]	]	PUNCT
ejpam-4854	367	24	t	t	NOUN
ejpam-4854	368	1	if	if	SCONJ
ejpam-4854	368	2	p	p	PROPN
ejpam-4854	368	3	=	=	NOUN
ejpam-4854	368	4	2	2	NUM
ejpam-4854	368	5	,	,	PUNCT
ejpam-4854	368	6	and	and	CCONJ
ejpam-4854	368	7	k	k	PROPN
ejpam-4854	368	8	=	=	SYM
ejpam-4854	368	9	3	3	NUM
ejpam-4854	368	10	,	,	PUNCT
ejpam-4854	368	11	we	we	PRON
ejpam-4854	368	12	obtain	obtain	VERB
ejpam-4854	368	13			NOUN
ejpam-4854	368	14	1	1	NUM
ejpam-4854	368	15	0	0	NUM
ejpam-4854	368	16	0	0	NUM
ejpam-4854	368	17	0	0	NUM
ejpam-4854	369	1	r	r	NOUN
ejpam-4854	369	2	1	1	NUM
ejpam-4854	369	3	0	0	NUM
ejpam-4854	369	4	0	0	NUM
ejpam-4854	369	5	r2	r2	NOUN
ejpam-4854	369	6	β	β	X
ejpam-4854	369	7	+	+	CCONJ
ejpam-4854	369	8	2r	2r	NUM
ejpam-4854	369	9	2	2	NUM
ejpam-4854	369	10	0	0	NUM
ejpam-4854	369	11	r3	r3	NOUN
ejpam-4854	369	12	β3	β3	NOUN
ejpam-4854	370	1	+	+	X
ejpam-4854	370	2	3βr	3βr	ADJ
ejpam-4854	370	3	+	+	CCONJ
ejpam-4854	370	4	3r2	3r2	NUM
ejpam-4854	370	5	6β	6β	NOUN
ejpam-4854	370	6	+	+	CCONJ
ejpam-4854	370	7	6r	6r	NUM
ejpam-4854	370	8	6	6	NUM
ejpam-4854	370	9			ADJ
ejpam-4854	370	10			NOUN
ejpam-4854	370	11	(	(	PUNCT
ejpam-4854	370	12	n+1	n+1	PROPN
ejpam-4854	370	13	2	2	NUM
ejpam-4854	370	14	)	)	PUNCT
ejpam-4854	370	15	β	β	X
ejpam-4854	370	16	(	(	PUNCT
ejpam-4854	370	17	n+1	n+1	PROPN
ejpam-4854	370	18	3	3	X
ejpam-4854	370	19	)	)	PUNCT
ejpam-4854	370	20	β2	β2	NOUN
ejpam-4854	370	21	(	(	PUNCT
ejpam-4854	370	22	n+1	n+1	PROPN
ejpam-4854	370	23	4	4	NUM
ejpam-4854	370	24	)	)	PUNCT
ejpam-4854	370	25	β3	β3	PROPN
ejpam-4854	370	26	(	(	PUNCT
ejpam-4854	370	27	n+1	n+1	PROPN
ejpam-4854	370	28	5	5	NUM
ejpam-4854	370	29	)	)	PUNCT
ejpam-4854	370	30			ADJ
ejpam-4854	370	31	=	=	NOUN
ejpam-4854	370	32			NOUN
ejpam-4854	370	33	1	1	NUM
ejpam-4854	370	34	2	2	NUM
ejpam-4854	370	35	n(n+	n(n+	NUM
ejpam-4854	370	36	1	1	NUM
ejpam-4854	370	37	)	)	SYM
ejpam-4854	370	38	1	1	NUM
ejpam-4854	370	39	6	6	NUM
ejpam-4854	370	40	n(n+	n(n+	NOUN
ejpam-4854	370	41	1)(βn−	1)(βn−	PRON
ejpam-4854	370	42	β	β	NOUN
ejpam-4854	370	43	+	+	NOUN
ejpam-4854	370	44	3r	3r	NUM
ejpam-4854	370	45	)	)	PUNCT
ejpam-4854	370	46	1	1	NUM
ejpam-4854	370	47	12	12	NUM
ejpam-4854	370	48	n(n+	n(n+	NOUN
ejpam-4854	370	49	1)(β2n2	1)(β2n2	NUM
ejpam-4854	370	50	−	−	NOUN
ejpam-4854	370	51	β2n+	β2n+	NOUN
ejpam-4854	370	52	4βnr	4βnr	NUM
ejpam-4854	371	1	+	+	NUM
ejpam-4854	372	1	6r2	6r2	NUM
ejpam-4854	372	2	)	)	PUNCT
ejpam-4854	372	3	1	1	NUM
ejpam-4854	372	4	60	60	NUM
ejpam-4854	372	5	n(n+	n(n+	NUM
ejpam-4854	372	6	1)(3β3n3	1)(3β3n3	NUM
ejpam-4854	372	7	−	−	PROPN
ejpam-4854	372	8	3β3n2	3β3n2	PRON
ejpam-4854	373	1	−	−	NOUN
ejpam-4854	373	2	2β3n+	2β3n+	NUM
ejpam-4854	373	3	2β3	2β3	NUM
ejpam-4854	374	1	+	+	CCONJ
ejpam-4854	374	2	15β2n2r	15β2n2r	NUM
ejpam-4854	375	1	−	−	NUM
ejpam-4854	375	2	15β2nr	15β2nr	NOUN
ejpam-4854	376	1	+	+	CCONJ
ejpam-4854	376	2	30βnr2	30βnr2	NUM
ejpam-4854	376	3	−	−	NUM
ejpam-4854	376	4	30βr2	30βr2	NOUN
ejpam-4854	376	5	+	+	CCONJ
ejpam-4854	376	6	30r3	30r3	NUM
ejpam-4854	376	7	)	)	PUNCT
ejpam-4854	376	8			NOUN
ejpam-4854	376	9	=	=	PUNCT
ejpam-4854	377	1	[	[	X
ejpam-4854	377	2	∗	∗	NOUN
ejpam-4854	377	3	,	,	PUNCT
ejpam-4854	377	4	z1	z1	NOUN
ejpam-4854	377	5	2,β	2,β	ADV
ejpam-4854	377	6	,	,	PUNCT
ejpam-4854	377	7	r(n	r(n	PROPN
ejpam-4854	377	8	)	)	PUNCT
ejpam-4854	377	9	,	,	PUNCT
ejpam-4854	377	10	z	z	NOUN
ejpam-4854	377	11	2	2	NUM
ejpam-4854	377	12	2,β	2,β	PRON
ejpam-4854	377	13	,	,	PUNCT
ejpam-4854	377	14	r(n	r(n	PROPN
ejpam-4854	377	15	)	)	PUNCT
ejpam-4854	377	16	,	,	PUNCT
ejpam-4854	377	17	z	z	NOUN
ejpam-4854	377	18	3	3	NUM
ejpam-4854	377	19	2,β	2,β	ADV
ejpam-4854	377	20	,	,	PUNCT
ejpam-4854	377	21	r(n	r(n	PROPN
ejpam-4854	377	22	)	)	PUNCT
ejpam-4854	377	23	]	]	PUNCT
ejpam-4854	377	24	t	t	PROPN
ejpam-4854	377	25	.	.	PUNCT
ejpam-4854	378	1	note	note	VERB
ejpam-4854	378	2	that	that	SCONJ
ejpam-4854	378	3	z1	z1	ADJ
ejpam-4854	378	4	2,m	2,m	NOUN
ejpam-4854	378	5	,	,	PUNCT
ejpam-4854	378	6	r(n	r(n	PROPN
ejpam-4854	378	7	)	)	PUNCT
ejpam-4854	378	8	,	,	PUNCT
ejpam-4854	378	9	k	k	PROPN
ejpam-4854	379	1	=	=	SYM
ejpam-4854	379	2	1	1	NUM
ejpam-4854	379	3	,	,	PUNCT
ejpam-4854	379	4	2	2	NUM
ejpam-4854	379	5	,	,	PUNCT
ejpam-4854	379	6	3	3	NUM
ejpam-4854	379	7	,	,	PUNCT
ejpam-4854	379	8	expresses	express	VERB
ejpam-4854	379	9	rk+(rk+(β+r)k)+	rk+(rk+(β+r)k)+	PROPN
ejpam-4854	379	10	·	·	PUNCT
ejpam-4854	379	11	·	·	PUNCT
ejpam-4854	380	1	·	·	PUNCT
ejpam-4854	380	2	+	+	PROPN
ejpam-4854	380	3	(	(	PUNCT
ejpam-4854	380	4	rk+(β+r)k+(2β+r)k)+	rk+(β+r)k+(2β+r)k)+	PROPN
ejpam-4854	380	5	·	·	PUNCT
ejpam-4854	380	6	·	·	PUNCT
ejpam-4854	380	7	·	·	PUNCT
ejpam-4854	380	8	+	+	ADJ
ejpam-4854	380	9	(	(	PUNCT
ejpam-4854	380	10	(	(	PUNCT
ejpam-4854	380	11	n−1)β+r)k	n−1)β+r)k	NOUN
ejpam-4854	380	12	)	)	PUNCT
ejpam-4854	380	13	=	=	PUNCT
ejpam-4854	380	14	n∑	n∑	NOUN
ejpam-4854	380	15	l=1	l=1	PROPN
ejpam-4854	380	16	l−1∑	l−1∑	ADJ
ejpam-4854	380	17	j=0	j=0	PROPN
ejpam-4854	380	18	(	(	PUNCT
ejpam-4854	380	19	jβ+r)k	jβ+r)k	PROPN
ejpam-4854	380	20	.	.	PUNCT
ejpam-4854	380	21	corollary	corollary	ADJ
ejpam-4854	380	22	3	3	NUM
ejpam-4854	380	23	.	.	PUNCT
ejpam-4854	381	1	for	for	ADP
ejpam-4854	381	2	each	each	DET
ejpam-4854	381	3	p	p	NOUN
ejpam-4854	381	4	=	=	SYM
ejpam-4854	381	5	1	1	NUM
ejpam-4854	381	6	,	,	PUNCT
ejpam-4854	381	7	2	2	NUM
ejpam-4854	381	8	,	,	PUNCT
ejpam-4854	381	9	.	.	PUNCT
ejpam-4854	381	10	.	.	PUNCT
ejpam-4854	381	11	.	.	PUNCT
ejpam-4854	382	1	,	,	PUNCT
ejpam-4854	382	2	n	n	CCONJ
ejpam-4854	382	3	,	,	PUNCT
ejpam-4854	382	4	we	we	PRON
ejpam-4854	382	5	have	have	VERB
ejpam-4854	382	6	np∑	np∑	NOUN
ejpam-4854	382	7	np−1=1	np−1=1	NOUN
ejpam-4854	382	8	np−1∑	np−1∑	NOUN
ejpam-4854	382	9	np−2=1	np−2=1	PUNCT
ejpam-4854	382	10	·	·	PUNCT
ejpam-4854	382	11	·	·	PUNCT
ejpam-4854	382	12	·	·	PUNCT
ejpam-4854	382	13	n1−1∑	n1−1∑	PUNCT
ejpam-4854	383	1	i=0	i=0	PROPN
ejpam-4854	383	2	(	(	PUNCT
ejpam-4854	383	3	iβ	iβ	ADP
ejpam-4854	383	4	+	+	CCONJ
ejpam-4854	383	5	r)k	r)k	X
ejpam-4854	383	6	=	=	PUNCT
ejpam-4854	383	7	k∑	k∑	NOUN
ejpam-4854	383	8	i=0	i=0	PROPN
ejpam-4854	383	9	i	i	NOUN
ejpam-4854	383	10	!	!	PUNCT
ejpam-4854	384	1	〈	〈	PROPN
ejpam-4854	384	2	k	k	X
ejpam-4854	384	3	i	i	PRON
ejpam-4854	384	4	〉	〉	NOUN
ejpam-4854	384	5	β	β	VERB
ejpam-4854	384	6	,	,	PUNCT
ejpam-4854	384	7	r	r	NOUN
ejpam-4854	384	8	(	(	PUNCT
ejpam-4854	384	9	n+	n+	X
ejpam-4854	384	10	p−	p−	NOUN
ejpam-4854	384	11	1	1	NUM
ejpam-4854	384	12	p+	p+	NOUN
ejpam-4854	384	13	i	i	NOUN
ejpam-4854	384	14	)	)	PUNCT
ejpam-4854	385	1	=	=	PUNCT
ejpam-4854	385	2	zk	zk	PROPN
ejpam-4854	385	3	p	p	PROPN
ejpam-4854	385	4	,	,	PUNCT
ejpam-4854	385	5	β	β	X
ejpam-4854	385	6	,	,	PUNCT
ejpam-4854	385	7	r(n	r(n	PROPN
ejpam-4854	385	8	)	)	PUNCT
ejpam-4854	385	9	,	,	PUNCT
ejpam-4854	385	10	where	where	SCONJ
ejpam-4854	385	11	np	np	ADV
ejpam-4854	385	12	=	=	PUNCT
ejpam-4854	385	13	n.	n.	NOUN
ejpam-4854	385	14	references	reference	NOUN
ejpam-4854	385	15	2322	2322	NUM
ejpam-4854	385	16	references	reference	NOUN
ejpam-4854	385	17	[	[	X
ejpam-4854	385	18	1	1	NUM
ejpam-4854	385	19	]	]	PUNCT
ejpam-4854	385	20	a.	a.	NOUN
ejpam-4854	385	21	bazsó	bazsó	NOUN
ejpam-4854	385	22	and	and	CCONJ
ejpam-4854	385	23	a.	a.	PROPN
ejpam-4854	385	24	pintér	pintér	PROPN
ejpam-4854	385	25	.	.	PUNCT
ejpam-4854	386	1	a	a	DET
ejpam-4854	386	2	refinement	refinement	NOUN
ejpam-4854	386	3	of	of	ADP
ejpam-4854	386	4	faulhaber	faulhaber	PROPN
ejpam-4854	386	5	’s	’s	PART
ejpam-4854	386	6	theorem	theorem	ADJ
ejpam-4854	386	7	concerning	concern	VERB
ejpam-4854	386	8	sums	sum	NOUN
ejpam-4854	386	9	of	of	ADP
ejpam-4854	386	10	powers	power	NOUN
ejpam-4854	386	11	of	of	ADP
ejpam-4854	386	12	natural	natural	ADJ
ejpam-4854	386	13	numbers	number	NOUN
ejpam-4854	386	14	.	.	PUNCT
ejpam-4854	387	1	applied	apply	VERB
ejpam-4854	387	2	mathematics	mathematics	NOUN
ejpam-4854	387	3	letters	letter	NOUN
ejpam-4854	387	4	.	.	PUNCT
ejpam-4854	387	5	,	,	PUNCT
ejpam-4854	387	6	25(3):486–489	25(3):486–489	PROPN
ejpam-4854	387	7	,	,	PUNCT
ejpam-4854	387	8	2012	2012	NUM
ejpam-4854	387	9	.	.	PUNCT
ejpam-4854	388	1	[	[	X
ejpam-4854	388	2	2	2	X
ejpam-4854	388	3	]	]	PUNCT
ejpam-4854	388	4	g.	g.	NOUN
ejpam-4854	388	5	call	call	PROPN
ejpam-4854	388	6	and	and	CCONJ
ejpam-4854	388	7	d.	d.	PROPN
ejpam-4854	388	8	velleman	velleman	PROPN
ejpam-4854	388	9	.	.	PUNCT
ejpam-4854	389	1	pascal	pascal	PROPN
ejpam-4854	389	2	’s	’s	PART
ejpam-4854	389	3	matrices	matrix	NOUN
ejpam-4854	389	4	.	.	PUNCT
ejpam-4854	390	1	the	the	DET
ejpam-4854	390	2	american	american	PROPN
ejpam-4854	390	3	mathematical	mathematical	PROPN
ejpam-4854	390	4	monthly	monthly	ADV
ejpam-4854	390	5	.	.	PUNCT
ejpam-4854	390	6	,	,	PUNCT
ejpam-4854	390	7	100(4):372	100(4):372	NUM
ejpam-4854	390	8	,	,	PUNCT
ejpam-4854	390	9	1993	1993	NUM
ejpam-4854	390	10	.	.	PUNCT
ejpam-4854	391	1	[	[	X
ejpam-4854	391	2	3	3	X
ejpam-4854	391	3	]	]	X
ejpam-4854	391	4	g.	g.	PROPN
ejpam-4854	391	5	cheon	cheon	PROPN
ejpam-4854	391	6	and	and	CCONJ
ejpam-4854	391	7	j.	j.	PROPN
ejpam-4854	391	8	kim	kim	PROPN
ejpam-4854	391	9	.	.	PUNCT
ejpam-4854	392	1	stirling	stirling	NOUN
ejpam-4854	392	2	matrix	matrix	NOUN
ejpam-4854	392	3	via	via	ADP
ejpam-4854	392	4	pascal	pascal	ADJ
ejpam-4854	392	5	matrix	matrix	NOUN
ejpam-4854	392	6	.	.	PUNCT
ejpam-4854	393	1	linear	linear	ADJ
ejpam-4854	393	2	algebra	algebra	NOUN
ejpam-4854	393	3	and	and	CCONJ
ejpam-4854	393	4	its	its	PRON
ejpam-4854	393	5	applications	application	NOUN
ejpam-4854	393	6	.	.	PUNCT
ejpam-4854	393	7	,	,	PUNCT
ejpam-4854	393	8	329(1	329(1	NUM
ejpam-4854	393	9	-	-	SYM
ejpam-4854	393	10	3):49–59	3):49–59	NUM
ejpam-4854	393	11	,	,	PUNCT
ejpam-4854	393	12	2001	2001	NUM
ejpam-4854	393	13	.	.	PUNCT
ejpam-4854	394	1	[	[	X
ejpam-4854	394	2	4	4	NUM
ejpam-4854	394	3	]	]	X
ejpam-4854	394	4	cb	cb	PROPN
ejpam-4854	394	5	corcino	corcino	PROPN
ejpam-4854	394	6	,	,	PUNCT
ejpam-4854	394	7	rb	rb	NOUN
ejpam-4854	394	8	corcino	corcino	NOUN
ejpam-4854	394	9	,	,	PUNCT
ejpam-4854	394	10	i	i	PRON
ejpam-4854	394	11	mező	mező	PROPN
ejpam-4854	394	12	,	,	PUNCT
ejpam-4854	394	13	and	and	CCONJ
ejpam-4854	394	14	jl	jl	PROPN
ejpam-4854	394	15	ramrez	ramrez	NOUN
ejpam-4854	394	16	.	.	PUNCT
ejpam-4854	395	1	some	some	DET
ejpam-4854	395	2	polynomials	polynomial	NOUN
ejpam-4854	395	3	associated	associate	VERB
ejpam-4854	395	4	with	with	ADP
ejpam-4854	395	5	the	the	DET
ejpam-4854	395	6	r	r	PROPN
ejpam-4854	395	7	-	-	PUNCT
ejpam-4854	395	8	whitney	whitney	NOUN
ejpam-4854	395	9	numbers	number	NOUN
ejpam-4854	395	10	.	.	PUNCT
ejpam-4854	396	1	proceedings	proceeding	NOUN
ejpam-4854	396	2	-	-	PUNCT
ejpam-4854	396	3	mathematical	mathematical	ADJ
ejpam-4854	396	4	sciences	science	NOUN
ejpam-4854	396	5	,	,	PUNCT
ejpam-4854	396	6	128(3):27	128(3):27	NUM
ejpam-4854	396	7	.	.	PUNCT
ejpam-4854	397	1	[	[	X
ejpam-4854	397	2	5	5	NUM
ejpam-4854	397	3	]	]	X
ejpam-4854	397	4	r.	r.	PROPN
ejpam-4854	397	5	corcino	corcino	PROPN
ejpam-4854	397	6	.	.	PUNCT
ejpam-4854	398	1	the	the	DET
ejpam-4854	398	2	(	(	PUNCT
ejpam-4854	398	3	r	r	NOUN
ejpam-4854	398	4	,	,	PUNCT
ejpam-4854	398	5	β)-stirling	β)-stirle	VERB
ejpam-4854	398	6	numbers	number	NOUN
ejpam-4854	398	7	.	.	PUNCT
ejpam-4854	399	1	mindanao	mindanao	PROPN
ejpam-4854	399	2	forum	forum	PROPN
ejpam-4854	399	3	,	,	PUNCT
ejpam-4854	399	4	14(2):91–100	14(2):91–100	NUM
ejpam-4854	399	5	,	,	PUNCT
ejpam-4854	399	6	2015	2015	NUM
ejpam-4854	399	7	.	.	PUNCT
ejpam-4854	400	1	[	[	X
ejpam-4854	400	2	6	6	NUM
ejpam-4854	400	3	]	]	X
ejpam-4854	400	4	r.	r.	PROPN
ejpam-4854	400	5	corcino	corcino	PROPN
ejpam-4854	400	6	and	and	CCONJ
ejpam-4854	400	7	r.	r.	PROPN
ejpam-4854	400	8	aldema	aldema	PROPN
ejpam-4854	400	9	.	.	PUNCT
ejpam-4854	401	1	some	some	DET
ejpam-4854	401	2	combinatorial	combinatorial	ADJ
ejpam-4854	401	3	and	and	CCONJ
ejpam-4854	401	4	statistical	statistical	ADJ
ejpam-4854	401	5	applications	application	NOUN
ejpam-4854	401	6	of	of	ADP
ejpam-4854	401	7	(	(	PUNCT
ejpam-4854	401	8	r	r	NOUN
ejpam-4854	401	9	,	,	PUNCT
ejpam-4854	401	10	β)stirling	β)stirle	VERB
ejpam-4854	401	11	numbers	number	NOUN
ejpam-4854	401	12	.	.	PUNCT
ejpam-4854	402	1	matimyás	matimyás	NOUN
ejpam-4854	402	2	matematika	matematika	NOUN
ejpam-4854	402	3	,	,	PUNCT
ejpam-4854	402	4	(	(	PUNCT
ejpam-4854	402	5	25	25	NUM
ejpam-4854	402	6	)	)	PUNCT
ejpam-4854	402	7	,	,	PUNCT
ejpam-4854	402	8	2002	2002	NUM
ejpam-4854	402	9	.	.	PUNCT
ejpam-4854	403	1	[	[	X
ejpam-4854	403	2	7	7	X
ejpam-4854	403	3	]	]	X
ejpam-4854	403	4	r.	r.	PROPN
ejpam-4854	403	5	corcino	corcino	PROPN
ejpam-4854	403	6	and	and	CCONJ
ejpam-4854	403	7	m.	m.	PROPN
ejpam-4854	403	8	montero	montero	PROPN
ejpam-4854	403	9	.	.	PUNCT
ejpam-4854	404	1	the	the	DET
ejpam-4854	404	2	(	(	PUNCT
ejpam-4854	404	3	r	r	NOUN
ejpam-4854	404	4	,	,	PUNCT
ejpam-4854	404	5	β)-stirling	β)-stirle	VERB
ejpam-4854	404	6	numbers	number	NOUN
ejpam-4854	404	7	in	in	ADP
ejpam-4854	404	8	the	the	DET
ejpam-4854	404	9	context	context	NOUN
ejpam-4854	404	10	of	of	ADP
ejpam-4854	404	11	0	0	NUM
ejpam-4854	404	12	-1	-1	PROPN
ejpam-4854	404	13	tableau	tableau	PROPN
ejpam-4854	404	14	.	.	PUNCT
ejpam-4854	405	1	journal	journal	PROPN
ejpam-4854	405	2	of	of	ADP
ejpam-4854	405	3	the	the	DET
ejpam-4854	405	4	mathematical	mathematical	ADJ
ejpam-4854	405	5	society	society	NOUN
ejpam-4854	405	6	of	of	ADP
ejpam-4854	405	7	the	the	DET
ejpam-4854	405	8	philippines	philippine	NOUN
ejpam-4854	405	9	,	,	PUNCT
ejpam-4854	405	10	32(1):42–52	32(1):42–52	NUM
ejpam-4854	405	11	,	,	PUNCT
ejpam-4854	405	12	2009	2009	NUM
ejpam-4854	405	13	.	.	PUNCT
ejpam-4854	406	1	[	[	X
ejpam-4854	406	2	8	8	NUM
ejpam-4854	406	3	]	]	X
ejpam-4854	406	4	r.b	r.b	PROPN
ejpam-4854	406	5	.	.	PROPN
ejpam-4854	406	6	corcino	corcino	PROPN
ejpam-4854	406	7	and	and	CCONJ
ejpam-4854	406	8	c.	c.	PROPN
ejpam-4854	406	9	barrientos	barrientos	PROPN
ejpam-4854	406	10	.	.	PUNCT
ejpam-4854	407	1	some	some	DET
ejpam-4854	407	2	theorems	theorem	NOUN
ejpam-4854	407	3	on	on	ADP
ejpam-4854	407	4	the	the	DET
ejpam-4854	407	5	q	q	NOUN
ejpam-4854	407	6	-	-	PUNCT
ejpam-4854	407	7	analogue	analogue	NOUN
ejpam-4854	407	8	of	of	ADP
ejpam-4854	407	9	the	the	DET
ejpam-4854	407	10	generalized	generalized	ADJ
ejpam-4854	407	11	stirling	stirling	NOUN
ejpam-4854	407	12	numbers	number	NOUN
ejpam-4854	407	13	.	.	PUNCT
ejpam-4854	408	1	bulletin	bulletin	NOUN
ejpam-4854	408	2	of	of	ADP
ejpam-4854	408	3	the	the	DET
ejpam-4854	408	4	malaysian	malaysian	PROPN
ejpam-4854	408	5	mathematical	mathematical	PROPN
ejpam-4854	408	6	sciences	sciences	PROPN
ejpam-4854	408	7	society	society	NOUN
ejpam-4854	408	8	,	,	PUNCT
ejpam-4854	408	9	34(3):487501	34(3):487501	NUM
ejpam-4854	408	10	,	,	PUNCT
ejpam-4854	408	11	2011	2011	NUM
ejpam-4854	408	12	.	.	PUNCT
ejpam-4854	409	1	[	[	X
ejpam-4854	409	2	9	9	NUM
ejpam-4854	409	3	]	]	PUNCT
ejpam-4854	409	4	rb	rb	NOUN
ejpam-4854	409	5	corcino	corcino	PROPN
ejpam-4854	409	6	,	,	PUNCT
ejpam-4854	409	7	cb	cb	PROPN
ejpam-4854	409	8	corcino	corcino	PROPN
ejpam-4854	409	9	,	,	PUNCT
ejpam-4854	409	10	and	and	CCONJ
ejpam-4854	409	11	r	r	NOUN
ejpam-4854	409	12	aldema	aldema	NOUN
ejpam-4854	409	13	.	.	PUNCT
ejpam-4854	410	1	asymptotic	asymptotic	ADJ
ejpam-4854	410	2	normality	normality	NOUN
ejpam-4854	410	3	of	of	ADP
ejpam-4854	410	4	the	the	DET
ejpam-4854	410	5	(	(	PUNCT
ejpam-4854	410	6	r	r	NOUN
ejpam-4854	410	7	,	,	PUNCT
ejpam-4854	410	8	β)-stirling	β)-stirle	VERB
ejpam-4854	410	9	numbers	number	NOUN
ejpam-4854	410	10	.	.	PUNCT
ejpam-4854	411	1	81:81–96	81:81–96	NUM
ejpam-4854	411	2	,	,	PUNCT
ejpam-4854	411	3	2006	2006	NUM
ejpam-4854	411	4	.	.	PUNCT
ejpam-4854	412	1	[	[	X
ejpam-4854	412	2	10	10	NUM
ejpam-4854	412	3	]	]	X
ejpam-4854	412	4	s.	s.	PROPN
ejpam-4854	412	5	getu	getu	PROPN
ejpam-4854	412	6	and	and	CCONJ
ejpam-4854	412	7	l.	l.	PROPN
ejpam-4854	412	8	shapiro	shapiro	PROPN
ejpam-4854	412	9	.	.	PUNCT
ejpam-4854	413	1	the	the	DET
ejpam-4854	413	2	riordan	riordan	PROPN
ejpam-4854	413	3	group	group	PROPN
ejpam-4854	413	4	.	.	PUNCT
ejpam-4854	414	1	discrete	discrete	ADJ
ejpam-4854	414	2	applied	applied	ADJ
ejpam-4854	414	3	mathematics	mathematic	NOUN
ejpam-4854	414	4	,	,	PUNCT
ejpam-4854	414	5	1991	1991	NUM
ejpam-4854	414	6	.	.	PUNCT
ejpam-4854	415	1	[	[	X
ejpam-4854	415	2	11	11	NUM
ejpam-4854	415	3	]	]	X
ejpam-4854	415	4	d.	d.	PROPN
ejpam-4854	415	5	kalman	kalman	PROPN
ejpam-4854	415	6	.	.	PUNCT
ejpam-4854	416	1	the	the	DET
ejpam-4854	416	2	generalized	generalized	ADJ
ejpam-4854	416	3	vandermonde	vandermonde	NOUN
ejpam-4854	416	4	matrix	matrix	NOUN
ejpam-4854	416	5	.	.	PUNCT
ejpam-4854	417	1	mathematics	mathematic	NOUN
ejpam-4854	417	2	magazine	magazine	NOUN
ejpam-4854	417	3	.	.	PUNCT
ejpam-4854	417	4	,	,	PUNCT
ejpam-4854	418	1	57(1):15	57(1):15	NUM
ejpam-4854	418	2	–	–	PUNCT
ejpam-4854	418	3	21	21	NUM
ejpam-4854	418	4	,	,	PUNCT
ejpam-4854	418	5	1984	1984	NUM
ejpam-4854	418	6	.	.	PUNCT
ejpam-4854	419	1	[	[	X
ejpam-4854	419	2	12	12	NUM
ejpam-4854	419	3	]	]	PUNCT
ejpam-4854	419	4	i.	i.	PROPN
ejpam-4854	419	5	mező	mező	PROPN
ejpam-4854	419	6	and	and	CCONJ
ejpam-4854	419	7	j.	j.	PROPN
ejpam-4854	419	8	ramı́rez	ramı́rez	PROPN
ejpam-4854	419	9	.	.	PUNCT
ejpam-4854	420	1	the	the	DET
ejpam-4854	420	2	linear	linear	PROPN
ejpam-4854	420	3	algebra	algebra	NOUN
ejpam-4854	420	4	of	of	ADP
ejpam-4854	420	5	the	the	DET
ejpam-4854	420	6	r	r	PROPN
ejpam-4854	420	7	-	-	PUNCT
ejpam-4854	420	8	whitney	whitney	NOUN
ejpam-4854	420	9	matices	matice	NOUN
ejpam-4854	420	10	.	.	PUNCT
ejpam-4854	421	1	integral	integral	ADJ
ejpam-4854	421	2	transforms	transform	NOUN
ejpam-4854	421	3	and	and	CCONJ
ejpam-4854	421	4	special	special	ADJ
ejpam-4854	421	5	functions	function	NOUN
ejpam-4854	421	6	.	.	PUNCT
ejpam-4854	421	7	,	,	PUNCT
ejpam-4854	421	8	26:213–225	26:213–225	NUM
ejpam-4854	421	9	,	,	PUNCT
ejpam-4854	421	10	2015	2015	NUM
ejpam-4854	421	11	.	.	PUNCT
ejpam-4854	422	1	[	[	X
ejpam-4854	422	2	13	13	NUM
ejpam-4854	422	3	]	]	PUNCT
ejpam-4854	422	4	z.	z.	PROPN
ejpam-4854	422	5	zhang	zhang	PROPN
ejpam-4854	422	6	.	.	PUNCT
ejpam-4854	423	1	the	the	DET
ejpam-4854	423	2	linear	linear	PROPN
ejpam-4854	423	3	algebra	algebra	NOUN
ejpam-4854	423	4	of	of	ADP
ejpam-4854	423	5	the	the	DET
ejpam-4854	423	6	generalized	generalized	ADJ
ejpam-4854	423	7	pascal	pascal	ADJ
ejpam-4854	423	8	matrix	matrix	NOUN
ejpam-4854	423	9	.	.	PUNCT
ejpam-4854	424	1	linear	linear	ADJ
ejpam-4854	424	2	algebra	algebra	NOUN
ejpam-4854	424	3	and	and	CCONJ
ejpam-4854	424	4	its	its	PRON
ejpam-4854	424	5	applications	application	NOUN
ejpam-4854	424	6	.	.	PUNCT
ejpam-4854	424	7	,	,	PUNCT
ejpam-4854	424	8	250:49–59	250:49–59	NUM
ejpam-4854	424	9	,	,	PUNCT
ejpam-4854	424	10	1997	1997	NUM
ejpam-4854	424	11	.	.	PUNCT