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A356460 Expansion of e.g.f. Product_{k>0} B(x^k)^k where B(x) = exp(exp(x)-1) = e.g.f. of Bell numbers.

%I A356460 #15 Aug 09 2022 11:19:08
%S A356460 1,1,6,35,303,2772,32903,410335,6051692,95183187,1675869175,
%T A356460 31437027030,644157830077,13976891765137,325719071472590,
%U A356460 8007861177420275,208953947981129027,5725964099963426924,165258064179632753563,4987477844227598529047
%N A356460 Expansion of e.g.f. Product_{k>0} B(x^k)^k where B(x) = exp(exp(x)-1) = e.g.f. of Bell numbers.
%F A356460 E.g.f.: Product_{k>0} exp(k * (exp(x^k)-1)).
%F A356460 a(0) = 1; a(n) = Sum_{k=1..n} A354863(k) * binomial(n-1,k-1) * a(n-k).
%o A356460 (PARI) my(N=20, x='x+O('x^N)); Vec(serlaplace(prod(k=1, N, exp(exp(x^k)-1)^k)))
%o A356460 (PARI) a354863(n) = n!*sumdiv(n, d, n/d/d!);
%o A356460 a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=sum(j=1, i, a354863(j)*binomial(i-1, j-1)*v[i-j+1])); v;
%Y A356460 Cf. A209902, A209903.
%Y A356460 Cf. A000110, A354863, A356461.
%K A356460 nonn
%O A356460 0,3
%A A356460 _Seiichi Manyama_, Aug 08 2022