A359761 a(n) = binomial(4*n, 2*n)*(2*n)!/(2^n*n!).

1, 6, 210, 13860, 1351350, 174594420, 28109701620, 5421156741000, 1218404977539750, 312723944235202500, 90252130306279441500, 28929910132721937339000, 10197793321784482911997500, 3920659309406065045704885000, 1632674555274097086889962825000, 732091270584905133761459330730000

Offset

0,2

Links

Table of n, a(n) for n=0..15.

Formula

a(n) = (2^(3*n)*Gamma(2*n + 1/2))/(sqrt(Pi)*Gamma(n + 1)).

a(n) = A359760(4*n, 2*n), the central terms of the triangle without the zeros.

From R. J. Mathar, Jan 25 2023: (Start)

a(n) = A001448(n)*A001147(n).

D-finite with recurrence n*a(n) - 2*(4*n-1)*(4*n-3)*a(n-1) = 0. (End)

From Stefano Spezia, Aug 24 2025: (Start)

E.g.f.: 2*EllipticK(8*sqrt(2*x)/(1 + 4*sqrt(2*x)))/(Pi*sqrt(1 + 4*sqrt(2*x))).

E.g.f.: hypergeom([1/2, 1/2], [1], 8*sqrt(2*x)/(1 + 4*sqrt(2*x)))/sqrt(1 + 4*sqrt(2*x)). (End)

a(n) ~ (32/e)^n * n^(n-1/2) / sqrt(Pi). - Amiram Eldar, Sep 05 2025

Maple

a := binomial(4*n, 2*n)*(2*n)!/(2^n*n!):
seq(a(n), n = 0..15);

Mathematica

a[n_] := Binomial[4*n, 2*n]*(2*n)!/(2^n*n!); Array[a, 20, 0] (* Amiram Eldar, Sep 05 2025 *)

Crossrefs

Cf. A001147, A001448, A359760.

Sequence in context: A183287 A087639 A028350 * A238685 A346017 A099788

Adjacent sequences: A359758 A359759 A359760 * A359762 A359763 A359764

Keyword

nonn,easy

Author

Peter Luschny, Jan 14 2023

Status

approved