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A355254 Expansion of e.g.f. exp(3*(exp(x) - 1) - x).

%I A355254 #14 Dec 04 2023 12:32:17
%S A355254 1,2,7,29,142,785,4813,32240,233449,1812161,14980768,131174939,
%T A355254 1211111629,11745451658,119255234371,1264050651953,13952113296766,
%U A355254 160006824960725,1902825936046105,23423342243273696,297982102750214605,3911917977005948453,52926119656555824520
%N A355254 Expansion of e.g.f. exp(3*(exp(x) - 1) - x).
%C A355254 Inverse binomial transform of A027710.
%C A355254 In general, if m >= 1 and e.g.f. = exp(m*exp(x) + r*x + s) then
%C A355254 a(n) ~ n^(n+r) * exp(n/LambertW(n/m) - n + s) / (m^r * sqrt(1 + LambertW(n/m)) * LambertW(n/m)^(n+r)).
%C A355254 Equivalently, a(n) ~ n! * (n/m)^r * exp(n/LambertW(n/m) + s) / (sqrt(2*Pi*n * (1 + LambertW(n/m))) * LambertW(n/m)^(n+r)).
%H A355254 Vaclav Kotesovec, <a href="/A355254/b355254.txt">Table of n, a(n) for n = 0..545</a>
%F A355254 a(n) ~ 3 * n^(n-1) * exp(n/LambertW(n/3) - n - 3) / (sqrt(1 + LambertW(n/3)) * LambertW(n/3)^(n-1)).
%F A355254 a(0) = 1; a(n) = -a(n-1) + 3 * Sum_{k=1..n} binomial(n-1,k-1) * a(n-k). - _Ilya Gutkovskiy_, Dec 04 2023
%t A355254 nmax = 25; CoefficientList[Series[Exp[3*Exp[x]-3-x], {x, 0, nmax}], x] * Range[0, nmax]!
%o A355254 (PARI) my(x='x+O('x^30)); Vec(serlaplace(exp(3*(exp(x) - 1) - x))) \\ _Michel Marcus_, Dec 04 2023
%Y A355254 Cf. A000296, A027710, A078940, A217924.
%K A355254 nonn
%O A355254 0,2
%A A355254 _Vaclav Kotesovec_, Jun 26 2022