A338979 a(n) = Sum_{k=0..n} n^k * binomial(n,k) * Catalan(k).

1, 2, 13, 199, 5073, 181776, 8413021, 478070020, 32238960193, 2517734880838, 223558608409101, 22248413487603887, 2453271411779452369, 296925818848604834448, 39138393489232585787037, 5581250331202285217569351, 856182695406472437496803585, 140595282922234695782098680030

Offset: 0

Author

Seiichi Manyama, Jan 31 2021

Keywords

• nonn
• changed

Links

• Seiichi Manyama, Table of n, a(n) for n = 0..322

Crossrefs

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

Programs

Maple

a := n -> hypergeom([1/2, -n], [2], -4*n):
seq(simplify(a(n)), n = 0..17);  # Peter Luschny, Aug 27 2025

Mathematica

A338979[n_] :=  Sum[n^k*Binomial[n, k]*(2*k)!/(k!*(k + 1)!), {k, 0, n}];
Join[{1}, Table[A338979[n], {n, 1, 17}]] (* Robert P. P. McKone, Jan 31 2021 *)
A338979[n_] := Hypergeometric2F1[1/2, -n, 2, -4*n]; Table[A338979[n], {n, 0, 17}]  (* Peter Luschny, Aug 27 2025 *)

PARI

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

Formula

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

a(n) ~ exp(1/4) * 4^n * n^(n - 3/2) / sqrt(Pi). - Vaclav Kotesovec, Feb 14 2021

a(n) = hypergeom([1/2, -n], [2], -4*n). - Peter Luschny, Aug 27 2025