A355380 - OEIS

%I #15 Jul 03 2022 04:45:25

%S 1,5,38,355,3879,48050,661163,9961745,162598044,2851150665,

%T 53350521523,1059447004560,22224898346989,490589320542305,

%U 11356591577861398,274886065370874775,6939205217774546339,182273695066097752170,4971724931587003394863,140559648864263508395965

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

%H Seiichi Manyama, <a href="/A355380/b355380.txt">Table of n, a(n) for n = 0..463</a>

%F a(n) = Sum_{k=0..n} binomial(n,k) * 3^k * 2^(n-k) * Bell(k) * Bell(n-k).

%F a(0) = 1; a(n) = Sum_{k=1..n} (3^k + 2^k) * binomial(n-1,k-1) * a(n-k). - _Seiichi Manyama_, Jun 30 2022

%F a(n) ~ exp(exp(3*z) + exp(2*z) - 2 - n) * (n/z)^(n + 1/2) / sqrt(3*(1 + 3*z)*exp(3*z) + 2*(1 + 2*z)*exp(2*z)), where z = LambertW(n)/3 - 1/(2 + 3/LambertW(n) + 9 * n^(1/3) * (1 + LambertW(n)) / (2*LambertW(n)^(4/3))). - _Vaclav Kotesovec_, Jul 03 2022

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

%t Table[Sum[Binomial[n,k] * 3^k * 2^(n-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(2*x) - 2))) \\ _Michel Marcus_, Jun 30 2022

%Y Cf. A143405, A355291, A355381.

%K nonn

%O 0,2

%A _Vaclav Kotesovec_, Jun 30 2022