A228156 Expansion of sqrt((1+4*x)/AGM(1+4*x,1-4*x)) where AGM denotes the arithmetic-geometric mean.

1, 2, 0, 8, 2, 68, 32, 720, 464, 8480, 6656, 106368, 95912, 1390928, 1392512, 18734144, 20371650, 257955716, 300101760, 3613109008, 4448177412, 51302395528, 66289160512, 736588435360, 992578330048, 10674012880512, 14924667774976, 155890890782720, 225244659392784, 2291995151532576, 3410654921389824

Offset

0,2

Comments

Convolution square is A092266.

Links

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

Formula

a(n) ~ 2^(2*n - 1/2) / (n*sqrt(Pi*log(n))) * (1 - (gamma + 3*log(2)) / (2*log(n)) + (3*gamma^2/8 + 9*gamma*log(2)/4 + 27*log(2)^2/8 - 1/16*Pi^2) / log(n)^2), where gamma is the Euler-Mascheroni constant A001620. - Vaclav Kotesovec, Sep 29 2019

Mathematica

CoefficientList[Series[Sqrt[2*(1 + 4*x)*EllipticK[1 - (1 + 4*x)^2/(1 - 4*x)^2] / (Pi*(1 - 4*x))], {x, 0, 30}], x] (* Vaclav Kotesovec, Sep 27 2019 *)

Prog

(PARI) Vec( 1/agm(1, (1-4*x)/(1+4*x)+O(x^66))^(1/2) ) \\ Joerg Arndt, Aug 14 2013

Crossrefs

Cf. A092266 (1+4*x)/AGM(1+4*x,1-4*x).

Cf. A081085 1/AGM(1,1-8*x), A053175 1/AGM(1,1-16*x), A090004 1/AGM(1,1-16*x)^(1/2), A089602 1/AGM(1,1-16*x)^(1/4).

Sequence in context: A206436 A146543 A179990 * A182524 A212832 A021052

Adjacent sequences: A228153 A228154 A228155 * A228157 A228158 A228159

Keyword

nonn

Author

Joerg Arndt, Aug 14 2013

Status

approved