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A354504 Expansion of e.g.f. ( Product_{k>0} (1 + x^k)^k )^exp(x).

%I A354504 #15 Aug 16 2022 10:19:49
%S A354504 1,1,6,48,402,4375,54595,777189,12284188,215999025,4132338673,
%T A354504 85640640877,1910121348674,45571124446445,1157169377895739,
%U A354504 31150000798832647,885481496002286200,26498034473000080321,832407848080194500301,27378188500890922864153
%N A354504 Expansion of e.g.f. ( Product_{k>0} (1 + x^k)^k )^exp(x).
%F A354504 a(0) = 1; a(n) = Sum_{k=1..n} A354508(k) * binomial(n-1,k-1) * a(n-k).
%o A354504 (PARI) my(N=20, x='x+O('x^N)); Vec(serlaplace(prod(k=1, N, (1+x^k)^k)^exp(x)))
%o A354504 (PARI) a354508(n) = n!*sum(k=1, n, sumdiv(k, d, (-1)^(k/d+1)*d^2)/(k*(n-k)!));
%o A354504 a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=sum(j=1, i, a354508(j)*binomial(i-1, j-1)*v[i-j+1])); v;
%Y A354504 Cf. A347915, A354503.
%Y A354504 Cf. A354508, A356394.
%K A354504 nonn
%O A354504 0,3
%A A354504 _Seiichi Manyama_, Aug 15 2022