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Expansion of e.g.f. exp(x^3/6 * (exp(x) - 1)).
7

%I #18 Oct 07 2023 11:40:51

%S 1,0,0,0,4,10,20,35,616,5124,29520,138765,942700,9369646,91711984,

%T 782281955,6539493520,62576274440,693828386976,7968383514969,

%U 89851862221140,1023732374445970,12384993316732960,160496534000858671,2163244034675904664,29653387436468336300

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

%H Winston de Greef, <a href="/A354001/b354001.txt">Table of n, a(n) for n = 0..529</a>

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

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

%t 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 *)

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

%o (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;

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

%Y Cf. A052506, A354000.

%Y Cf. A351493, A353999.

%K nonn

%O 0,5

%A _Seiichi Manyama_, May 13 2022