%I #26 Oct 04 2023 15:07:59
%S 1,0,-4,-12,16,400,2208,-448,-131840,-1357056,-4820480,71120896,
%T 1537308672,14006460416,3075702784,-2224350781440,-41354996154368,
%U -359660395495424,1675436608585728,121894823709900800,2317859245604208640,20543311167964053504
%N Expansion of e.g.f. exp(x * (1 - exp(2*x))).
%H Seiichi Manyama, <a href="/A356812/b356812.txt">Table of n, a(n) for n = 0..499</a>
%F G.f.: Sum_{k>=0} (-x)^k / (1 - (2*k+1)*x)^(k+1).
%F a(n) = Sum_{k=0..n} (-1)^k * (2*k+1)^(n-k) * binomial(n,k).
%F a(n) = n! * Sum_{k=0..floor(n/2)} (-1)^k * 2^(n-k) * Stirling2(n-k,k)/(n-k)!.
%t With[{nn=30},CoefficientList[Series[Exp[x(1-Exp[2x])],{x,0,nn}],x] Range[0,nn]!] (* _Harvey P. Dale_, Oct 04 2023 *)
%o (PARI) my(N=30, x='x+O('x^N)); Vec(serlaplace(exp(x*(1-exp(2*x)))))
%o (PARI) my(N=30, x='x+O('x^N)); Vec(sum(k=0, N, (-x)^k/(1-(2*k+1)*x)^(k+1)))
%o (PARI) a(n) = sum(k=0, n, (-1)^k*(2*k+1)^(n-k)*binomial(n, k));
%o (PARI) a(n) = n!*sum(k=0, n\2, (-1)^k*2^(n-k)*stirling(n-k, k, 2)/(n-k)!);
%Y Cf. A292893, A356813.
%Y Cf. A351736, A356815, A356819.
%K sign
%O 0,3
%A _Seiichi Manyama_, Aug 29 2022