|
%I #27 May 07 2022 05:53:25
%S 1,0,0,0,24,240,2040,17640,202776,3066336,52446720,933636000,
%T 17416490784,350580364992,7719355635264,184232862777600,
%U 4691944607751936,126358891201529856,3591751011211717632,107772466927523060736,3408777017097439186944
%N Expansion of e.g.f. 1/(1 - log(1 - x)^4).
%H Seiichi Manyama, <a href="/A353119/b353119.txt">Table of n, a(n) for n = 0..418</a>
%F a(0) = 1; a(n) = 24 * Sum_{k=1..n} binomial(n,k) * |Stirling1(k,4)| * a(n-k).
%F a(n) = Sum_{k=0..floor(n/4)} (4*k)! * |Stirling1(n,4*k)|.
%F a(n) ~ sqrt(2*Pi) * n^(n + 1/2) / (4 * (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)^4)))
%o (PARI) a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=24*sum(j=1, i, binomial(i, j)*abs(stirling(j, 4, 1))*v[i-j+1])); v;
%o (PARI) a(n) = sum(k=0, n\4, (4*k)!*abs(stirling(n, 4*k, 1)));
%Y Cf. A052811, A353118, A353200.
%Y Cf. A346923.
%K nonn
%O 0,5
%A _Seiichi Manyama_, May 06 2022
|
