A060691 Expansion of AGM(1,1-8x) (where AGM denotes the arithmetic-geometric mean).

1, -4, -4, -16, -84, -496, -3120, -20416, -137300, -942384, -6572336, -46432960, -331580272, -2389352256, -17351364160, -126851634432, -932823545428, -6895102385072, -51199649648048, -381738099675840, -2856639909232112

Offset: 0

Author

Roland Bacher, Apr 20 2001

Keywords

• sign

Links

• Vaclav Kotesovec, Table of n, a(n) for n = 0..1000

Crossrefs

Cf. A081085.

Programs

Mathematica

CoefficientList[Series[1/Hypergeometric2F1[1/2, 1/2, 1, 16*x*(1 - 4*x)], {x, 0, 20}], x] (* Vaclav Kotesovec, Jan 13 2019 *)
CoefficientList[Series[Pi*(1 - 4*x)/(2*EllipticK[1/(1 - 1/(4*x))^2]), {x, 0, 20}], x] (* Vaclav Kotesovec, Jan 13 2019 *)

PARI

a(n)=if(n<0,0,polcoeff(agm(1,1-8*x+x*O(x^n)),n))

Formula

G.f.: AGM(1, 1-8x).

a(n) ~ -Pi * 2^(3*n-1) / (n * log(n)^2) * (1 - (2*gamma + 4*log(2))/log(n) + (3*gamma^2 + 12*log(2)*gamma + 12*log(2)^2 - Pi^2/2) / log(n)^2), where gamma is the Euler-Mascheroni constant A001620. - Vaclav Kotesovec, Sep 29 2019

Extensions

Edited by Michael Somos, Jul 19 2002