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

%I #24 Feb 15 2021 04:56:21

%S 1,1,9,75,3675,59535,2401245,57972915,13043905875,418854310875,

%T 30241281245175,1212400457192925,213786613951685775,

%U 10278202593831046875,1070401384414690453125,60013837619516978071875,57673297952355815927071875,3694483615889146090857721875

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

%C The denominators are given in A123854.

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

%C 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.

%F 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).

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

%e 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, ...

%t With[{n = 20}, Numerator[CoefficientList[Series[2 EllipticK[x]/Pi, {x, 0, n}], x] Range[0, n]!]] (* _Jan Mangaldan_, Jan 04 2017 *)

%t Numerator[Table[Gamma[n + 1/2]^2/(Pi Gamma[n + 1]), {n, 0, 20}]] (* _Li Han_, Feb 05 2021 *)

%Y Cf. A038534/A056982, A123854.

%K nonn,easy,frac

%O 0,3

%A _Wolfdieter Lang_, Jul 07 2016