%I #13 Dec 04 2023 12:32:50

%S 1,5,27,157,979,6517,46107,345261,2726243,22623525,196712171,

%T 1787356765,16929897395,166808851541,1706299041211,18088031239437,

%U 198392625389315,2248104026019461,26283054263021963,316637825898555069,3926250785070282579,50056384077880370101

%N Expansion of e.g.f. exp(2*(exp(x) - 1) + 3*x).

%C Binomial transform of A355247.

%H Vaclav Kotesovec, <a href="/A355252/b355252.txt">Table of n, a(n) for n = 0..550</a>

%F a(n) ~ n^(n+3) * exp(n/LambertW(n/2) - n - 2) / (8 * 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 = 25; 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, A035009, A355247, A355253.

%K nonn

%O 0,2

%A _Vaclav Kotesovec_, Jun 26 2022