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Expansion of e.g.f. exp(x^3 * (exp(x) - 1)).
4

%I #28 Jul 10 2022 08:08:11

%S 1,0,0,0,24,60,120,210,20496,181944,1059120,4990590,100458600,

%T 1634594676,18436740504,164378216730,2124284725920,38171412643440,

%U 631390188466656,8760417873485814,124649582165430840,2167585391936047020,41833303600025220360

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

%H Seiichi Manyama, <a href="/A292891/b292891.txt">Table of n, a(n) for n = 0..200</a>

%F From _Seiichi Manyama_, Jul 09 2022: (Start)

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

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

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

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

%o (PARI) a(n) = n!*sum(k=0, n\4, stirling(n-3*k, k, 2)/(n-3*k)!); \\ _Seiichi Manyama_, Jul 09 2022

%o (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

%Y Column k=3 of A292892.

%Y Cf. A292889, A354001.

%K nonn

%O 0,5

%A _Seiichi Manyama_, Sep 26 2017