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%I #21 Aug 29 2022 16:35:51
%S 1,0,4,12,80,560,4512,40768,407808,4453632,52605440,667234304,
%T 9032423424,129822564352,1972450443264,31559866736640,530043925495808,
%U 9317136303718400,170976603113127936,3268020569256755200,64928967058257346560,1338431135849666052096
%N Expansion of e.g.f. exp( x * (exp(2 * x) - 1) ).
%H Seiichi Manyama, <a href="/A351736/b351736.txt">Table of n, a(n) for n = 0..488</a>
%F a(n) = n! * Sum_{k=0..floor(n/2)} 2^(n-k) * Stirling2(n-k,k)/(n-k)!.
%F From _Seiichi Manyama_, Aug 29 2022: (Start)
%F a(n) = Sum_{k=0..n} (2*k-1)^(n-k) * binomial(n,k).
%F G.f.: Sum_{k>=0} x^k / (1 - (2*k-1)*x)^(k+1). (End)
%o (PARI) my(N=30, x='x+O('x^N)); Vec(serlaplace(exp(x*(exp(2*x)-1))))
%o (PARI) a(n) = n!*sum(k=0, n\2, 2^(n-k)*stirling(n-k, k, 2)/(n-k)!);
%o (PARI) a(n) = sum(k=0, n, (2*k-1)^(n-k)*binomial(n, k)); \\ _Seiichi Manyama_, Aug 29 2022
%o (PARI) my(N=30, x='x+O('x^N)); Vec(sum(k=0, N, x^k/(1-(2*k-1)*x)^(k+1))) \\ _Seiichi Manyama_, Aug 29 2022
%Y Cf. A052506, A351737.
%Y Cf. A053491, A351733.
%K nonn
%O 0,3
%A _Seiichi Manyama_, May 20 2022
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