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

1, 0, 0, 0, 24, 240, 1560, 8400, 81144, 1638000, 31058520, 482499600, 6905646264, 114015261360, 2456232531480, 59734751403600, 1427946773067384, 33377481440110320, 818549745973204440, 22338800420915168400, 667566534457962216504, 20735588176755396824880

Offset

0,5

Links

Seiichi Manyama, Table of n, a(n) for n = 0..424

Formula

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

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

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

a(n) ~ n! / (8 * log(2)^(n+1)). - Vaclav Kotesovec, May 08 2022

Mathematica

With[{nn=30}, CoefficientList[Series[1/(1-(Exp[x]-1)^4), {x, 0, nn}], x] Range[0, nn]!] (* Harvey P. Dale, Apr 05 2023 *)

Prog

(PARI) my(N=30, x='x+O('x^N)); Vec(serlaplace(1/(1-(exp(x)-1)^4)))
(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)))
(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;
(PARI) a(n) = sum(k=0, n\4, (4*k)!*stirling(n, 4*k, 2));

Crossrefs

Cf. A000670, A052841, A353774.

Cf. A346895, A353119, A353665.

Sequence in context: A056269 A000919 A353665 * A268966 A014340 A052753

Adjacent sequences: A353772 A353773 A353774 * A353776 A353777 A353778

Keyword

nonn

Author

Seiichi Manyama, May 07 2022

Status

approved