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A354291 Expansion of e.g.f. exp(f(x) - 1) where f(x) = 1/(4 - 3*exp(x)) = e.g.f. for A032033.

%I A354291 #14 May 23 2022 11:27:26
%S A354291 1,3,30,435,8211,190056,5196099,163541055,5815620696,230350071189,
%T A354291 10048990989747,478467217544322,24678559536271581,1370217125170670367,
%U A354291 81457311857722336614,5160975525978898855143,347090708803947931122807,24690132231344937537382560
%N A354291 Expansion of e.g.f. exp(f(x) - 1) where f(x) = 1/(4 - 3*exp(x)) = e.g.f. for A032033.
%F A354291 a(0) = 1; a(n) = Sum_{k=1..n} A032033(k) * binomial(n-1,k-1) * a(n-k).
%F A354291 a(n) = Sum_{k=0..n} 3^k * A000262(k) * Stirling2(n,k).
%F A354291 a(n) ~ exp(-7/8 - n + 1/(8*log(4/3)) + sqrt(n/log(4/3))) * n^(n - 1/4) / (2*log(4/3)^(n + 1/4)). - _Vaclav Kotesovec_, May 23 2022
%o A354291 (PARI) my(N=20, x='x+O('x^N)); Vec(serlaplace(exp(3*(exp(x)-1)/(4-3*exp(x)))))
%o A354291 (PARI) a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=sum(j=1, i, sum(k=0, j, 3^k*k!*stirling(j, k, 2))*binomial(i-1, j-1)*v[i-j+1])); v;
%Y A354291 Cf. A075729, A354290.
%Y A354291 Cf. A032033, A354287, A354289.
%K A354291 nonn
%O A354291 0,2
%A A354291 _Seiichi Manyama_, May 23 2022