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A355378 Expansion of e.g.f. exp(exp(3*x) - exp(x)).

%I A355378 #18 Jul 10 2022 07:12:52
%S A355378 1,2,12,82,688,6754,75096,928386,12591392,185384130,2938319144,
%T A355378 49799613538,897495547184,17118975292514,344206910941624,
%U A355378 7270287035936706,160826794265399360,3716047107259486082,89472755268582494792,2240097688067896960674,58207872357772581544272
%N A355378 Expansion of e.g.f. exp(exp(3*x) - exp(x)).
%H A355378 Seiichi Manyama, <a href="/A355378/b355378.txt">Table of n, a(n) for n = 0..467</a>
%F A355378 a(n) = Sum_{k=0..n} binomial(n,k) * 3^k * Bell(k) * Bell(n-k, -1).
%F A355378 a(0) = 1; a(n) = Sum_{k=1..n} (3^k - 1) * binomial(n-1,k-1) * a(n-k). - _Seiichi Manyama_, Jun 30 2022
%F A355378 a(n) ~ exp(exp(3*z) - exp(z) - n) * (n/z)^(n + 1/2) / sqrt(3*(1 + 3*z)*exp(3*z) - (1 + z)*exp(z)), where z = LambertW(n)/3 - 1/(1 + 3/LambertW(n) - 9 * n^(2/3) * (1 + LambertW(n)) / LambertW(n)^(5/3)). - _Vaclav Kotesovec_, Jul 03 2022
%F A355378 a(n) ~ (3*n/LambertW(n))^n * exp(n/LambertW(n) - (n/LambertW(n))^(1/3) - n) / sqrt(1 + LambertW(n)). - _Vaclav Kotesovec_, Jul 10 2022
%t A355378 nmax = 20; CoefficientList[Series[Exp[Exp[3*x] - Exp[x]], {x, 0, nmax}], x] * Range[0, nmax]!
%t A355378 Table[Sum[Binomial[n,k] * 3^k * BellB[k] * BellB[n-k, -1], {k, 0, n}], {n, 0, 20}]
%o A355378 (PARI) my(x='x+O('x^25)); Vec(serlaplace(exp(exp(3*x) - exp(x)))) \\ _Michel Marcus_, Jun 30 2022
%Y A355378 Cf. A143405, A355291, A355379, A355381.
%K A355378 nonn
%O A355378 0,2
%A A355378 _Vaclav Kotesovec_, Jun 30 2022