A353200 - OEIS

%I #40 May 07 2022 05:51:36

%S 1,0,0,0,0,120,1800,21000,235200,2693880,35947800,609615600,

%T 12504927600,281242996320,6545492073120,155873050569600,

%U 3849612346944000,100588974863402880,2818516832681523840,84728757269204858880,2706516690047188416000

%N Expansion of e.g.f. 1/(1 + log(1 - x)^5).

%H Seiichi Manyama, <a href="/A353200/b353200.txt">Table of n, a(n) for n = 0..418</a>

%F a(0) = 1; a(n) = 120 * Sum_{k=1..n} binomial(n,k) * |Stirling1(k,5)| * a(n-k).

%F a(n) = Sum_{k=0..floor(n/5)} (5*k)! * |Stirling1(n,5*k)|.

%F a(n) ~ sqrt(2*Pi) * n^(n + 1/2) / (5 * (exp(1) - 1)^(n+1)). - _Vaclav Kotesovec_, May 07 2022

%o (PARI) my(N=30, x='x+O('x^N)); Vec(serlaplace(1/(1+log(1-x)^5)))

%o (PARI) a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=120*sum(j=1, i, binomial(i, j)*abs(stirling(j, 5, 1))*v[i-j+1])); v;

%o (PARI) a(n) = sum(k=0, n\5, (5*k)!*abs(stirling(n, 5*k, 1)));

%Y Cf. A052811, A353118, A353119.

%Y Cf. A346924.

%K nonn

%O 0,6

%A _Seiichi Manyama_, May 06 2022