A387488 a(n) = [x^n] hypergeom([1/2, 1/2], [1], 256*x).

1, 64, 9216, 1638400, 321126400, 66588770304, 14323984367616, 3161800304492544, 711405068510822400, 162446273421928038400, 37531587011402253991936, 8754475304213365427011584, 2058274415968386804839612416, 487165542241038297003458560000, 115965283361213687678537564160000

Offset

0,2

Links

Table of n, a(n) for n=0..14.

mpmath, Elliptic functions, 3.0 documentation.

Formula

a(n) = [x^n] 2*EllipticK((16*sqrt(x))^m)/Pi where m = 1 if the branch cuts follow the Maple conventions and m = 2 if they follow the Mathematica conventions.

a(n) ~ 2^(8*n) / (Pi*n). - Vaclav Kotesovec, Sep 05 2025

Maple

gf := hypergeom([1/2, 1/2], [1], 256*x): ser := series(gf, x, 20): seq(coeff(ser, x, n), n = 0..14);
# Or:
ge := 2*EllipticK(16*sqrt(x))/Pi: ser := series(ge, x, 20): seq(coeff(ser, x, n), n = 0..14);

Mathematica

s[x_] := HypergeometricPFQ[{1/2, 1/2}, {1}, 256*x]; CoefficientList[Series[s[x], {x, 0, 14}], x]
(* Or: *)
gf[x_] := 2*EllipticK[256*x]/Pi; CoefficientList[Series[gf[x], {x, 0, 14}], x]
(* or *)
CoefficientList[Series[1/ArithmeticGeometricMean[1, Sqrt[1 - 256*x]], {x, 0, 14}], x] (* Vaclav Kotesovec, Sep 05 2025 *)

Prog

(PARI) N=20; x='x+O('x^N); Vec(1/agm(1, sqrt(1 - 256*x))) \\ Vaclav Kotesovec, Sep 05 2025

Crossrefs

Cf. A036910, A359761.

Sequence in context: A239647 A222140 A093264 * A264146 A220731 A239403

Adjacent sequences: A387485 A387486 A387487 * A387489 A387490 A387491

Keyword

nonn

Author

Peter Luschny, Sep 05 2025

Status

approved