A371399 a(n) = 2^n * Sum_{k=0..n} binomial(k + n, k) * binomial(2*n - k, n) * (-1/2)^k.

1, 2, 12, 60, 340, 1932, 11256, 66264, 394020, 2359500, 14211912, 86004360, 522502344, 3184844600, 19467675120, 119288938800, 732508344516, 4506518476620, 27771180181800, 171393806476200, 1059200506065240, 6553715347503720, 40595235803924880, 251709010315822800

Offset: 0

Author

Peter Luschny, Mar 21 2024

Keywords

• nonn

Crossrefs

Cf. A371400.

Programs

Maple

seq((2^n*add(binomial(k+n, k)*binomial(2*n-k, n)*(-1/2)^k, k=0..n)), n=0..23);

Mathematica

a[n_] := 2^n Binomial[2 n, n] Hypergeometric2F1[-n, 1 + n, -2 n, -1/2];
Table[a[n], {n, 0, 23}]

Python

from math import comb
def A371399(n): return sum(comb(k+n,k)*comb((n<<1)-k,n)*(-1 if k&1 else 1)<Chai Wah Wu, Mar 22 2024

Formula

a(n) = 2^n * Sum_{k=0..n} A371400(n, k) * (-1/2)^k.

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