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Expansion of e.g.f. 1/(1 - (exp(x) - 1)^4).
6

%I #19 Apr 05 2023 10:21:16

%S 1,0,0,0,24,240,1560,8400,81144,1638000,31058520,482499600,6905646264,

%T 114015261360,2456232531480,59734751403600,1427946773067384,

%U 33377481440110320,818549745973204440,22338800420915168400,667566534457962216504,20735588176755396824880

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

%H Seiichi Manyama, <a href="/A353775/b353775.txt">Table of n, a(n) for n = 0..424</a>

%F G.f.: Sum_{k>=0} (4*k)! * x^(4*k)/Product_{j=1..4*k} (1 - j * x).

%F a(0) = 1; a(n) = 24 * Sum_{k=1..n} binomial(n,k) * Stirling2(k,4) * a(n-k).

%F a(n) = Sum_{k=0..floor(n/4)} (4*k)! * Stirling2(n,4*k).

%F a(n) ~ n! / (8 * log(2)^(n+1)). - _Vaclav Kotesovec_, May 08 2022

%t 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 (PARI) my(N=30, x='x+O('x^N)); Vec(serlaplace(1/(1-(exp(x)-1)^4)))

%o (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 (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 (PARI) a(n) = sum(k=0, n\4, (4*k)!*stirling(n, 4*k, 2));

%Y Cf. A000670, A052841, A353774.

%Y Cf. A346895, A353119, A353665.

%K nonn

%O 0,5

%A _Seiichi Manyama_, May 07 2022