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%I #12 Dec 04 2023 12:32:22
%S 1,-1,3,-5,19,-29,171,-69,2339,5139,57563,303403,2397011,17237507,
%T 139011211,1151110299,10076637827,91903924979,874688607035,
%U 8656097294091,88932728790195,946748093175523,10426787247224043,118620906668843131,1392128306377939427,16833088095308098003
%N Expansion of e.g.f. exp(2*(exp(x) - 1) - 3*x).
%C Inverse binomial transform of A194689.
%H Vaclav Kotesovec, <a href="/A355253/b355253.txt">Table of n, a(n) for n = 0..555</a>
%F a(n) ~ 8 * n^(n-3) * exp(n/LambertW(n/2) - n - 2) / (sqrt(1 + LambertW(n/2)) * LambertW(n/2)^(n-3)).
%F a(0) = 1; a(n) = -3 * a(n-1) + 2 * Sum_{k=1..n} binomial(n-1,k-1) * a(n-k). - _Ilya Gutkovskiy_, Dec 04 2023
%t nmax = 30; CoefficientList[Series[Exp[2*Exp[x]-2-3*x], {x, 0, nmax}], x] * Range[0, nmax]!
%o (PARI) my(x='x+O('x^30)); Vec(serlaplace(exp(2*(exp(x) - 1) - 3*x))) \\ _Michel Marcus_, Dec 04 2023
%Y Cf. A001861, A217923, A194689, A355252.
%K sign
%O 0,3
%A _Vaclav Kotesovec_, Jun 26 2022