A358109 a(n) = 16^n * Sum_{k=0..n} binomial(1/2, k)^2 * binomial(n, k).

1, 20, 388, 7376, 138340, 2572880, 47652240, 882388800, 16402291620, 307411770320, 5837516987920, 112918906836800, 2237687548230160, 45677390764531520, 964818477552462400, 21148251536958233856, 481370160754727691300, 11360399185583414128848, 277079154699775861823376

Offset: 0

Author

Peter Luschny, Nov 12 2022

Keywords

• nonn

Comments

Belongs to the family of Apéry-like sequences.

Crossrefs

Cf. A358108, A143583.

Programs

Maple

a := n -> 16^n*add(binomial(1/2, k)^2*binomial(n, k), k = 0..n):
seq(a(n), n = 0..18);

Mathematica

a[n_] := 16^n * HypergeometricPFQ[{-1/2, -1/2, -n}, {1, 1}, -1]; Array[a, 19, 0] (* Amiram Eldar, Nov 12 2022 *)

Python

from sympy import binomial, S
def A358109(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^3). - Vaclav Kotesovec, Nov 12 2022