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%I #39 Aug 17 2022 02:36:54
%S 1,0,2,9,44,390,2754,32760,310064,4244184,54887400,818615160,
%T 12909921672,225872515440,4045885572624,79360837887240,
%U 1649832369335040,35666417240193600,822291935260976064,19830352438530840960,501144432316767688320,13229590606682042436480
%N Expansion of e.g.f. ( Product_{k>0} 1/(1-x^k) )^x.
%F a(0) = 1, a(1) = 0; a(n) = Sum_{k=2..n} k! * sigma(k-1)/(k-1) * binomial(n-1,k-1) * a(n-k).
%t nmax = 20; CoefficientList[Series[Product[1/(1 - x^k)^x, {k, 1, nmax}], {x, 0, nmax}], x] * Range[0, nmax]! (* _Vaclav Kotesovec_, Aug 17 2022 *)
%o (PARI) my(N=30, x='x+O('x^N)); Vec(serlaplace(1/prod(k=1, N, 1-x^k)^x))
%o (PARI) a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=sum(j=2, i, j!*sigma(j-1)/(j-1)*binomial(i-1, j-1)*v[i-j+1])); v;
%Y Cf. A355064, A356554.
%Y Cf. A000203, A053529, A066166, A356335, A356565.
%K nonn
%O 0,3
%A _Seiichi Manyama_, Aug 12 2022