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A257635 Triangle with n-th row polynomial equal to Product_{k = 1..n} (x + n + k). 2
1, 2, 1, 12, 7, 1, 120, 74, 15, 1, 1680, 1066, 251, 26, 1, 30240, 19524, 5000, 635, 40, 1, 665280, 434568, 117454, 16815, 1345, 57, 1, 17297280, 11393808, 3197348, 495544, 45815, 2527, 77, 1, 518918400, 343976400, 99236556, 16275700, 1659889, 107800, 4354, 100, 1 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

The row polynomials are a Sheffer sequence. For the associated polynomial sequence of binomial type see A038455.

LINKS

Table of n, a(n) for n=0..44.

R. Sprugnoli, An Introduction to Mathematical Methods in Combinatorics, Section 5.6 CreateSpace Independent Publishing Platform 2006, ISBN-13: 978-1502925244

Wikipedia, Sheffer sequence

FORMULA

E.g.f. A(x,t) = B(t)*C(t)^x = 1 + (2 + x)*t + (3 + x)*(4 + x)*t^2/2! + (4 + x)*(5 + x)*(6 + x)*t^3/3! + ..., where B(t) = 1/sqrt(1 - 4*t) is the o.g.f. for A000984 and C(t) = (1 - sqrt(1 - 4*t))/(2*t) is the o.g.f. for A000108.

n-th row polynomial: n!*binomial(2*n + x,n).

EXAMPLE

Triangle begins

1,

2, 1,

12, 7, 1,

120, 74, 15, 1,

1680, 1066, 251, 26, 1,

30240, 19524, 5000, 635, 40, 1,

665280, 434568, 117454, 16815, 1345, 57, 1,

...

MAPLE

#A257635

seq(seq(coeff(product(n + x + k, k = 1 .. n), x, i), i = 0..n), n = 0..8);

CROSSREFS

A001813 (column 0), A005449 (first subdiagonal), A098118 (column 1). Cf. A000108, A000984, A038455, A092932.

Sequence in context: A132875 A050139 A010255 * A085752 A074966 A128413

Adjacent sequences:  A257632 A257633 A257634 * A257636 A257637 A257638

KEYWORD

nonn,tabl,easy

AUTHOR

Peter Bala, Nov 05 2015

STATUS

approved

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Last modified October 21 20:44 EDT 2019. Contains 328315 sequences. (Running on oeis4.)