A338979 - OEIS

%I #30 Feb 14 2021 08:05:05

%S 1,2,13,199,5073,181776,8413021,478070020,32238960193,2517734880838,

%T 223558608409101,22248413487603887,2453271411779452369,

%U 296925818848604834448,39138393489232585787037

%N a(n) = Sum_{k=0..n} n^k * binomial(n,k) * Catalan(k).

%H Seiichi Manyama, <a href="/A338979/b338979.txt">Table of n, a(n) for n = 0..322</a>

%F a(n) = n! * [x^n] exp((2*n+1)*x) * (BesselI(0,2*n*x) - BesselI(1,2*n*x)). - _Ilya Gutkovskiy_, Feb 02 2021

%F a(n) ~ exp(1/4) * 4^n * n^(n - 3/2) / sqrt(Pi). - _Vaclav Kotesovec_, Feb 14 2021

%t A338979[n_] :=  Sum[n^k*Binomial[n, k]*(2*k)!/(k!*(k + 1)!), {k, 0, n}]; Join[{1}, Table[A338979[n], {n, 1, 14}]] (* _Robert P. P. McKone_, Jan 31 2021 *)

%o (PARI) {a(n) = sum(k=0, n, n^k*binomial(n, k)*(2*k)!/(k!*(k+1)!))}

%Y Cf. A000108, A007317, A162326, A337167, A339001.

%K nonn

%O 0,2

%A _Seiichi Manyama_, Jan 31 2021