A300100 Expansion of sqrt(agm(1, 1 - 8*x)) in powers of x.

1, -2, -4, -16, -82, -476, -2968, -19360, -130220, -895592, -6264656, -44411968, -318300080, -2302042400, -16777460032, -123084642048, -908175062994, -6734680013532, -50164119638328, -375134475461088, -2815268948389212, -21195313970398536

Offset: 0

Author

Seiichi Manyama, Feb 24 2018

Keywords

• sign

Links

• Seiichi Manyama, Table of n, a(n) for n = 0..500
• Eric Weisstein's World of Mathematics, Arithmetic-geometric mean
• Wikipedia, Arithmetic-geometric mean

Crossrefs

Cf. A060691, A089603.

Programs

Mathematica

CoefficientList[Series[Sqrt[Pi*(1 - 8*x) / (2*EllipticK[1 - 1/(1 - 8*x)^2])], {x, 0, 25}], x] (* Vaclav Kotesovec, Sep 28 2019 *)

Formula

Convolution inverse of A089603.

a(n) ~ -sqrt(Pi) * 2^(3*n - 3/2) / (n * log(n)^(3/2)) * (1 - 3*(gamma/2 + log(2)) / log(n) + (15*gamma^2/8 + 15*log(2)*gamma/2 + 15*log(2)^2/2 - 5*Pi^2/16) / log(n)^2), where gamma is the Euler-Mascheroni constant A001620. - Vaclav Kotesovec, Sep 29 2019