A130189 Numerators of z-sequence for the Sheffer matrix (triangle) A094816 (coefficients of Poisson-Charlier polynomials).

1, -1, 5, -7, 68, -167, 2057, -4637, 75703, -39941, 676360, -902547, 602501827, -432761746, 2438757091, -8997865117, 346824403906, -1857709421899, 325976550837563, -282728710837871, 39928855264303811, -16874802689368067, 162083496666375118, -3212329557624761759

Offset: 0

Author

Wolfdieter Lang, Jun 01 2007

Keywords

• sign
• frac
• easy

Comments

The denominators are given in A130190.

This z-sequence is useful for the recurrence for S(n,m=0):= A094816(n,0) (first column): S(n,0) = n*Sum_{j=0..n-1} z(j)*S(n-1,j), n >= 1, S(0,0)=1.

See the W. Lang link under A006232 with a summary on a- and z-sequences for Sheffer matrices.

Examples

			Rationals z(n): [1, -1/2, 5/6, -7/4, 68/15, -167/12, 2057/42, -4637/24, ...].
Recurrence from z(n) sequence for S(n,0)= A094816(n,0) for n=4: 1 = S(4,0) = 4*(1*1 - (1/2)*8 + (5/6)*6 - (7/4)*1) with the 3rd row [1,8,6,1] of A094816.
		

Links

• G. C. Greubel, Table of n, a(n) for n = 0..569
• W. Lang, Rationals, z-sequence.

Crossrefs

Cf. A027641/A027642 (Bernoulli numbers) provide the a-sequence for the Sheffer matrix A094816.

Programs

Maple

seq(numer((-1)^n*add(Stirling2(n,k)/(k+1),k=0..n)),n=0..20); # Peter Luschny, Apr 28 2009

Mathematica

Table[(-1)^n*Numerator[Sum[StirlingS2[n, k]/(k + 1), {k, 0, n}]], {n, 0, 50}] (* G. C. Greubel, Jul 10 2018 *)

PARI

a(n) = (-1)^n*numerator(sum(k=0, n, stirling(n, k, 2)/(k+1))); \\ Michel Marcus, Jan 15 2015

Formula

E.g.f. for rationals z(n)=a(n)/A130190(n) (in lowest terms): (1-exp(-h(x)))/h(x) with h(x):=1-exp(-x).

Numerator of (-1)^n Sum_{k=0..n} A048993(n,k)/(k+1). - Peter Luschny, Apr 28 2009