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A228156
Expansion of sqrt((1+4*x)/AGM(1+4*x,1-4*x)) where AGM denotes the arithmetic-geometric mean.
1
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
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
KEYWORD
nonn
AUTHOR
Joerg Arndt, Aug 14 2013
STATUS
approved