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%I #25 Aug 06 2021 05:07:43
%S 1,-1,5,-7,68,-167,2057,-4637,75703,-39941,676360,-902547,602501827,
%T -432761746,2438757091,-8997865117,346824403906,-1857709421899,
%U 325976550837563,-282728710837871,39928855264303811,-16874802689368067,162083496666375118,-3212329557624761759
%N Numerators of z-sequence for the Sheffer matrix (triangle) A094816 (coefficients of Poisson-Charlier polynomials).
%C The denominators are given in A130190.
%C 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.
%C See the W. Lang link under A006232 with a summary on a- and z-sequences for Sheffer matrices.
%H G. C. Greubel, <a href="/A130189/b130189.txt">Table of n, a(n) for n = 0..569</a>
%H W. Lang, <a href="/A130189/a130189.txt">Rationals, z-sequence. </a>
%F 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).
%F Numerator of (-1)^n Sum_{k=0..n} A048993(n,k)/(k+1). - _Peter Luschny_, Apr 28 2009
%e Rationals z(n): [1, -1/2, 5/6, -7/4, 68/15, -167/12, 2057/42, -4637/24, ...].
%e 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.
%p seq(numer((-1)^n*add(Stirling2(n,k)/(k+1),k=0..n)),n=0..20); # _Peter Luschny_, Apr 28 2009
%t Table[(-1)^n*Numerator[Sum[StirlingS2[n, k]/(k + 1), {k, 0, n}]], {n, 0, 50}] (* _G. C. Greubel_, Jul 10 2018 *)
%o (PARI) a(n) = (-1)^n*numerator(sum(k=0, n, stirling(n, k, 2)/(k+1))); \\ _Michel Marcus_, Jan 15 2015
%Y Cf. A027641/A027642 (Bernoulli numbers) provide the a-sequence for the Sheffer matrix A094816.
%K sign,frac,easy
%O 0,3
%A _Wolfdieter Lang_, Jun 01 2007
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