A358108 a(n) = 16^n * Sum_{k=0..n} binomial(-1/2, k)^2 * binomial(n, k).

1, 20, 420, 9296, 216868, 5313360, 135866640, 3599688000, 98122746660, 2735243498960, 77595234251920, 2231860533960000, 64904359322352400, 1904342118510144320, 56285527873777258560, 1673824975976543421696, 50036226313229526706980, 1502471400349641645458640

Offset: 0

Author

Peter Luschny, Nov 12 2022

Keywords

• nonn

Comments

Belongs to the family of Apéry-like sequences.

Crossrefs

Cf. A143583.

Programs

Maple

a := n -> 16^n*add(binomial(-1/2, k)^2*binomial(n, k), k = 0..n):
seq(a(n), n = 0..17);

Mathematica

a[n_] := 16^n * HypergeometricPFQ[{1/2, 1/2, -n}, {1, 1}, -1]; Array[a, 18, 0] (* Amiram Eldar, Nov 12 2022 *)

Python

from sympy import binomial, S
def A358108(n): return (1<<(n<<2))*sum(binomial(-S.Half,k)**2*binomial(n,k) for k in range(n+1)) # Chai Wah Wu, Nov 13 2022

Formula

a(n) = 16^n * hypergeom([1/2, 1/2, -n], [1, 1], -1).

a(n) ~ 2^(5*n + 1) / (Pi*n). - Vaclav Kotesovec, Nov 12 2022