%I #14 Aug 16 2022 10:20:03
%S 1,1,3,13,54,291,1778,12167,82869,655100,5658257,51691806,454932679,
%T 4527660281,48270581011,553646849053,5561424579562,72988254250439,
%U 1010390962699396,12295679951427509,67360732923382327,1515500302797716376,45199587363022824107,1001538050395504921200,-699211952404047871075
%N Expansion of e.g.f. ( Product_{k>0} (1 + x^k)^(1/k!) )^exp(x).
%F a(0) = 1; a(n) = Sum_{k=1..n} A354509(k) * binomial(n-1,k-1) * a(n-k).
%o (PARI) my(N=30, x='x+O('x^N)); Vec(serlaplace(prod(k=1, N, (1+x^k)^(1/k!))^exp(x)))
%o (PARI) a354509(n) = n!*sum(k=1, n, sumdiv(k, d, (-1)^(d+1)/(d*(k/d)!))/(n-k)!);
%o a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=sum(j=1, i, a354509(j)*binomial(i-1, j-1)*v[i-j+1])); v;
%Y Cf. A354509, A356402.
%K sign
%O 0,3
%A _Seiichi Manyama_, Aug 15 2022