cp's OEIS Frontend

A355379 Expansion of e.g.f. exp(exp(3*x) + exp(x) - 2).

%I A355379 #17 Jul 10 2022 07:13:59
%S A355379 1,4,26,212,2046,22588,278942,3792916,56128254,895795692,15307847614,
%T A355379 278435732484,5364073445278,108994074306268,2327475127169182,
%U A355379 52069279762495220,1217024509006768574,29647115491635327180,751085909757123127294,19750410883486281805028
%N A355379 Expansion of e.g.f. exp(exp(3*x) + exp(x) - 2).
%H A355379 Seiichi Manyama, <a href="/A355379/b355379.txt">Table of n, a(n) for n = 0..466</a>
%F A355379 a(n) = Sum_{k=0..n} binomial(n,k) * 3^k * Bell(k) * Bell(n-k).
%F A355379 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 A355379 a(n) ~ exp(exp(3*z) + exp(z) - 2 - 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 A355379 a(n) ~ (3*n/LambertW(n))^n * exp(n/LambertW(n) + (n/LambertW(n))^(1/3) - n - 2) / sqrt(1 + LambertW(n)). - _Vaclav Kotesovec_, Jul 10 2022
%t A355379 nmax = 20; CoefficientList[Series[Exp[Exp[3*x] + Exp[x] - 2], {x, 0, nmax}], x] * Range[0, nmax]!
%t A355379 Table[Sum[Binomial[n,k] * 3^k * BellB[k] * BellB[n-k], {k, 0, n}], {n, 0, 20}]
%o A355379 (PARI) my(x='x+O('x^25)); Vec(serlaplace(exp(exp(3*x) + exp(x) - 2))) \\ _Michel Marcus_, Jun 30 2022
%Y A355379 Cf. A143405, A355291, A355378.
%K A355379 nonn
%O A355379 0,2
%A A355379 _Vaclav Kotesovec_, Jun 30 2022