A355379 - OEIS

%I #17 Jul 10 2022 07:13:59

%S 1,4,26,212,2046,22588,278942,3792916,56128254,895795692,15307847614,

%T 278435732484,5364073445278,108994074306268,2327475127169182,

%U 52069279762495220,1217024509006768574,29647115491635327180,751085909757123127294,19750410883486281805028

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

%H Seiichi Manyama, <a href="/A355379/b355379.txt">Table of n, a(n) for n = 0..466</a>

%F a(n) = Sum_{k=0..n} binomial(n,k) * 3^k * Bell(k) * Bell(n-k).

%F 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 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 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 nmax = 20; CoefficientList[Series[Exp[Exp[3*x] + Exp[x] - 2], {x, 0, nmax}], x] * Range[0, nmax]!

%t Table[Sum[Binomial[n,k] * 3^k * BellB[k] * BellB[n-k], {k, 0, n}], {n, 0, 20}]

%o (PARI) my(x='x+O('x^25)); Vec(serlaplace(exp(exp(3*x) + exp(x) - 2))) \\ _Michel Marcus_, Jun 30 2022

%Y Cf. A143405, A355291, A355378.

%K nonn

%O 0,2

%A _Vaclav Kotesovec_, Jun 30 2022