A354001 Expansion of e.g.f. exp(x^3/6 * (exp(x) - 1)).

1, 0, 0, 0, 4, 10, 20, 35, 616, 5124, 29520, 138765, 942700, 9369646, 91711984, 782281955, 6539493520, 62576274440, 693828386976, 7968383514969, 89851862221140, 1023732374445970, 12384993316732960, 160496534000858671, 2163244034675904664, 29653387436468336300

Offset: 0

Author

Seiichi Manyama, May 13 2022

Keywords

• nonn

Links

• Winston de Greef, Table of n, a(n) for n = 0..529

Crossrefs

Cf. A052506, A354000.

Cf. A351493, A353999.

Programs

Mathematica

With[{nn=30},CoefficientList[Series[Exp[x^3/6 (Exp[x]-1)],{x,0,nn}],x] Range[0,nn]!] (* Harvey P. Dale, Oct 07 2023 *)

PARI

my(N=30, x='x+O('x^N)); Vec(serlaplace(exp(x^3/6*(exp(x)-1))))

PARI

a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=(i-1)!/6*sum(j=4, i, j/(j-3)!*v[i-j+1]/(i-j)!)); v;

PARI

a(n) = n!*sum(k=0, n\4, stirling(n-3*k, k, 2)/(6^k*(n-3*k)!));

Formula

a(0) = 1; a(n) = ((n-1)!/6) * Sum_{k=4..n} k/(k-3)! * a(n-k)/(n-k)!.

a(n) = n! * Sum_{k=0..floor(n/4)} Stirling2(n-3*k,k)/(6^k * (n-3*k)!).