OFFSET
0,2
LINKS
Vaclav Kotesovec, Table of n, a(n) for n = 0..825
Vaclav Kotesovec, Graph - the asymptotic ratio (30000 terms)
Eric Weisstein's MathWorld, Complete Elliptic Integral of the First Kind.
Eric Weisstein's MathWorld, Complete Elliptic Integral of the Second Kind.
FORMULA
a(n) ~ -2^(4*n+1) / (n * log(n)^2) * (1 - (2*gamma + 8*log(2)) / log(n) + (3*gamma^2 + 24*log(2)*gamma + 48*log(2)^2 - Pi^2/2) / log(n)^2 + (-4*gamma^3 + 2*gamma*Pi^2 - 48*gamma^2*log(2) + 8*Pi^2*log(2) - 192*gamma*log(2)^2 - 256*log(2)^3 - 8*Zeta(3)) / log(n)^3 + (5*gamma^4 - 5*gamma^2*Pi^2 + Pi^4/12 + 80*gamma^3*log(2) - 40*gamma*Pi^2*log(2) + 480*gamma^2*log(2)^2 - 80*Pi^2*log(2)^2 + 1280*gamma*log(2)^3 + 1280*log(2)^4 + 40*gamma*Zeta(3) + 160*log(2)*Zeta(3)) / log(n)^4), where gamma is the Euler-Mascheroni constant A001620.
MAPLE
seq(coeff(series(EllipticE(4*sqrt(x))/EllipticK(4*sqrt(x)), x, 21), x, n), n = 0..20);
MATHEMATICA
CoefficientList[Series[EllipticE[16*x]/EllipticK[16*x], {x, 0, 20}], x]
CROSSREFS
KEYWORD
sign
AUTHOR
Vaclav Kotesovec, Oct 04 2019
STATUS
approved