A089603 G.f.: sqrt(1/agm(1, 1-8*x)) = sqrt(o.g.f. for A081085).

1, 2, 8, 40, 226, 1380, 8880, 59280, 406416, 2842400, 20186752, 145119616, 1053575336, 7711639760, 56834201280, 421327859520, 3139306406850, 23494847031300, 176526280319120, 1330929290036560, 10065855468854980, 76341682531733960, 580460500453098080

Offset: 0

Author

N. J. A. Sloane, Dec 31 2003

Keywords

• nonn

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. A081085, A089602.

Programs

Mathematica

CoefficientList[Series[Sqrt[2*EllipticK[1/(1 - 1/(4*x))^2]/(Pi*(1 - 4*x))], {x, 0, 25}], x] (* Vaclav Kotesovec, Sep 26 2019 *)
nmax = 25; CoefficientList[Series[Sqrt[Hypergeometric2F1[1/2, 1/2, 1, 16*x*(1 - 4*x)]], {x, 0, nmax}], x] (* Vaclav Kotesovec, Sep 26 2019 *)

PARI

Vec( 1/agm(1,1-8*x+O(x^66))^(1/2) ) \\ Joerg Arndt, Aug 14 2013

Formula

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