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A353775 Expansion of e.g.f. 1/(1 - (exp(x) - 1)^4).

%I A353775 #19 Apr 05 2023 10:21:16
%S A353775 1,0,0,0,24,240,1560,8400,81144,1638000,31058520,482499600,6905646264,
%T A353775 114015261360,2456232531480,59734751403600,1427946773067384,
%U A353775 33377481440110320,818549745973204440,22338800420915168400,667566534457962216504,20735588176755396824880
%N A353775 Expansion of e.g.f. 1/(1 - (exp(x) - 1)^4).
%H A353775 Seiichi Manyama, <a href="/A353775/b353775.txt">Table of n, a(n) for n = 0..424</a>
%F A353775 G.f.: Sum_{k>=0} (4*k)! * x^(4*k)/Product_{j=1..4*k} (1 - j * x).
%F A353775 a(0) = 1; a(n) = 24 * Sum_{k=1..n} binomial(n,k) * Stirling2(k,4) * a(n-k).
%F A353775 a(n) = Sum_{k=0..floor(n/4)} (4*k)! * Stirling2(n,4*k).
%F A353775 a(n) ~ n! / (8 * log(2)^(n+1)). - _Vaclav Kotesovec_, May 08 2022
%t A353775 With[{nn=30},CoefficientList[Series[1/(1-(Exp[x]-1)^4),{x,0,nn}],x] Range[0,nn]!] (* _Harvey P. Dale_, Apr 05 2023 *)
%o A353775 (PARI) my(N=30, x='x+O('x^N)); Vec(serlaplace(1/(1-(exp(x)-1)^4)))
%o A353775 (PARI) my(N=30, x='x+O('x^N)); Vec(sum(k=0, N, (4*k)!*x^(4*k)/prod(j=1, 4*k, 1-j*x)))
%o A353775 (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)*stirling(j, 4, 2)*v[i-j+1])); v;
%o A353775 (PARI) a(n) = sum(k=0, n\4, (4*k)!*stirling(n, 4*k, 2));
%Y A353775 Cf. A000670, A052841, A353774.
%Y A353775 Cf. A346895, A353119, A353665.
%K A353775 nonn
%O A353775 0,5
%A A353775 _Seiichi Manyama_, May 07 2022