A353229 Expansion of e.g.f. (1 - x)^(-x^3).

1, 0, 0, 0, 24, 60, 240, 1260, 28224, 241920, 2181600, 21621600, 315342720, 4358914560, 61607407680, 912518006400, 15142006978560, 265601118182400, 4877947688140800, 93691850626483200, 1901787789077452800, 40548028309147699200, 904101131200045363200

Offset: 0

Author

Seiichi Manyama, May 01 2022

Keywords

• nonn

Links

• Seiichi Manyama, Table of n, a(n) for n = 0..449

Crossrefs

Cf. A066166, A351493, A351504, A353228.

Programs

Mathematica

With[{nn=30},CoefficientList[Series[(1-x)^-x^3,{x,0,nn}],x] Range[0,nn]!] (* Harvey P. Dale, Jun 20 2024 *)

PARI

my(N=30, x='x+O('x^N)); Vec(serlaplace((1-x)^(-x^3)))

PARI

my(N=30, x='x+O('x^N)); Vec(serlaplace(exp(-x^3*log(1-x))))

PARI

a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=(i-1)!*sum(j=4, i, j/(j-3)*v[i-j+1]/(i-j)!)); v;

PARI

a(n) = n!*sum(k=0, n\4, abs(stirling(n-3*k, k, 1))/(n-3*k)!);

Formula

a(0) = 1; a(n) = (n-1)! * Sum_{k=4..n} k/(k-3) * a(n-k)/(n-k)!.

a(n) = n! * Sum_{k=0..floor(n/4)} |Stirling1(n-3*k,k)|/(n-3*k)!.

a(n) ~ sqrt(2*Pi) * n^(n + 1/2) / exp(n). - Vaclav Kotesovec, May 04 2022