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A354000 Expansion of e.g.f. exp(x^2/2 * (exp(x) - 1)).

%I A354000 #16 Sep 19 2023 01:50:22
%S A354000 1,0,0,3,6,10,105,651,2968,18936,152505,1164295,9109056,80012868,
%T A354000 756041377,7387199925,75535791360,816560002576,9254683835073,
%U A354000 109135702334619,1338613513677280,17079079303721820,226148006163689841,3100114305453613393,43935964285680790368
%N A354000 Expansion of e.g.f. exp(x^2/2 * (exp(x) - 1)).
%H A354000 Winston de Greef, <a href="/A354000/b354000.txt">Table of n, a(n) for n = 0..529</a>
%F A354000 a(0) = 1; a(n) = ((n-1)!/2) * Sum_{k=3..n} k/(k-2)! * a(n-k)/(n-k)!.
%F A354000 a(n) = n! * Sum_{k=0..floor(n/3)} Stirling2(n-2*k,k)/(2^k * (n-2*k)!).
%o A354000 (PARI) my(N=30, x='x+O('x^N)); Vec(serlaplace(exp(x^2/2*(exp(x)-1))))
%o A354000 (PARI) a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=(i-1)!/2*sum(j=3, i, j/(j-2)!*v[i-j+1]/(i-j)!)); v;
%o A354000 (PARI) a(n) = n!*sum(k=0, n\3, stirling(n-2*k, k, 2)/(2^k*(n-2*k)!));
%Y A354000 Cf. A052506, A354001.
%Y A354000 Cf. A353998, A351492.
%K A354000 nonn
%O A354000 0,4
%A A354000 _Seiichi Manyama_, May 13 2022