A292891 Expansion of e.g.f. exp(x^3 * (exp(x) - 1)).

1, 0, 0, 0, 24, 60, 120, 210, 20496, 181944, 1059120, 4990590, 100458600, 1634594676, 18436740504, 164378216730, 2124284725920, 38171412643440, 631390188466656, 8760417873485814, 124649582165430840, 2167585391936047020, 41833303600025220360

Offset: 0

Author

Seiichi Manyama, Sep 26 2017

Keywords

• nonn

Links

• Seiichi Manyama, Table of n, a(n) for n = 0..200

Crossrefs

Column k=3 of A292892.

Cf. A292889, A354001.

Programs

Mathematica

With[{nn=30},CoefficientList[Series[Exp[x^3 (Exp[x]-1)],{x,0,nn}],x] Range[0,nn]!] (* Harvey P. Dale, Feb 21 2022 *)

PARI

x='x+O('x^66); Vec(serlaplace(exp(x^3*(exp(x)-1))))

PARI

a(n) = n!*sum(k=0, n\4, stirling(n-3*k, k, 2)/(n-3*k)!); \\ Seiichi Manyama, Jul 09 2022

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; \\ Seiichi Manyama, Jul 09 2022

Formula

From Seiichi Manyama, Jul 09 2022: (Start)

a(n) = n! * Sum_{k=0..floor(n/4)} Stirling2(n-3*k,k)/(n-3*k)!.

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