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

0,3

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.

Links

Table of n, a(n) for n=0..17.

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).

Example

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, ...

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 *)

Crossrefs

Cf. A038534/A056982, A123854.

Sequence in context: A190916 A161736 A226180 * A255931 A319957 A056339

Adjacent sequences: A274654 A274655 A274656 * A274658 A274659 A274660

Keyword

nonn,easy,frac

Author

Wolfdieter Lang, Jul 07 2016

Status

approved