A274657 Numerators of the coefficients of z^n/n! for the expansion of hypergeometric([1/2,1/2],[1];z).

1, 1, 9, 75, 3675, 59535, 2401245, 57972915, 13043905875, 418854310875, 30241281245175, 1212400457192925, 213786613951685775, 10278202593831046875, 1070401384414690453125, 60013837619516978071875, 57673297952355815927071875, 3694483615889146090857721875

Offset: 0

Author

Wolfdieter Lang, Jul 07 2016

Keywords

• nonn
• easy
• frac

Comments

The denominators are given in A123854.

The main entry is A038534 (with A056982) where comments and references are given.

The complete elliptic integral of the first kind K = K(k) is (Pi/2)*hypergeometric([1/2,1/2],[1];k^2). This is also the real quarter period K of elliptic functions.

Examples

			The first rationals r(n) are: 1, 1/4, 9/32, 75/128, 3675/2048, 59535/8192, 2401245/65536, 57972915/262144, 13043905875/8388608, 418854310875/33554432, 30241281245175/268435456, ...
		

Crossrefs

Cf. A038534/A056982, A123854.

Programs

Mathematica

With[{n = 20}, Numerator[CoefficientList[Series[2 EllipticK[x]/Pi, {x, 0, n}], x] Range[0, n]!]] (* Jan Mangaldan, Jan 04 2017 *)
Numerator[Table[Gamma[n + 1/2]^2/(Pi Gamma[n + 1]), {n, 0, 20}]] (* Li Han, Feb 05 2021 *)

Formula

a(n) = numerator(r(n)) with the rationals (in lowest terms) r(n) = (risefac(1/2,n)^2)/n! = ((2*n)!^2)/((n!^3)*2^(4*n)), with the rising factorial risefac (Pochhammer symbol).

E.g.f. for r(n) is hypergeometric([1/2,1/2],[1];z).