A351734 Expansion of e.g.f. exp( 3 * x * (exp(x) - 1) ).

1, 0, 6, 9, 120, 555, 5148, 39711, 378528, 3715011, 39838260, 452684463, 5463506304, 69553644771, 930940368036, 13054086036855, 191222363275968, 2918620069099395, 46309955947643124, 762335523354333855, 12995722456718984160, 229045407317491457763

Offset: 0

Author

Seiichi Manyama, May 20 2022

Keywords

• nonn

Crossrefs

Cf. A052506, A351733.

Cf. A053490, A351737.

Programs

Mathematica

With[{nn=30},CoefficientList[Series[Exp[3x (Exp[x]-1)],{x,0,nn}],x] Range[0,nn]!] (* Harvey P. Dale, Apr 02 2025 *)

PARI

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

PARI

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

Formula

a(n) = n! * Sum_{k=0..floor(n/2)} 3^k * Stirling2(n-k,k)/(n-k)!.