A339001 a(n) = (-1)^n * Sum_{k=0..n} (-n)^k * binomial(n,k) * Catalan(k).

1, 0, 5, 89, 2481, 93274, 4450645, 258297570, 17689681345, 1397903887808, 125286890408901, 12562851683433765, 1393925069404093105, 169595051215441936902, 22454465186157134883285

Offset: 0

Author

Seiichi Manyama, Jan 31 2021

Keywords

• nonn

Links

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

Crossrefs

Cf. A000108, A005043, A337168, A337169, A338979.

Programs

Mathematica

a[0] = 1; a[n_] := (-1)^n * Sum[(-n)^k * Binomial[n, k] * CatalanNumber[k], {k, 0, n}]; Array[a, 15, 0] (* Amiram Eldar, Feb 01 2021 *)

PARI

{a(n) = (-1)^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 01 2021

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