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%I #8 Oct 05 2019 08:33:10
%S 1,-8,-16,-128,-1312,-15104,-186112,-2398208,-31898176,-434421248,
%T -6025687552,-84808699904,-1207939190272,-17375932633088,
%U -252046328713216,-3682284573851648,-54130292542567552,-800036763837307904,-11880834659028677632,-177181827571092267008
%N G.f.: E(4*sqrt(x)) / K(4*sqrt(x)), where E(), K() are complete elliptic integrals.
%H Vaclav Kotesovec, <a href="/A328127/b328127.txt">Table of n, a(n) for n = 0..825</a>
%H Vaclav Kotesovec, <a href="/A328127/a328127.jpg">Graph - the asymptotic ratio (30000 terms)</a>
%H Eric Weisstein's MathWorld, <a href="http://mathworld.wolfram.com/CompleteEllipticIntegraloftheFirstKind.html">Complete Elliptic Integral of the First Kind</a>.
%H Eric Weisstein's MathWorld, <a href="http://mathworld.wolfram.com/CompleteEllipticIntegraloftheSecondKind.html">Complete Elliptic Integral of the Second Kind</a>.
%F 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.
%p seq(coeff(series(EllipticE(4*sqrt(x))/EllipticK(4*sqrt(x)), x, 21), x, n), n = 0..20);
%t CoefficientList[Series[EllipticE[16*x]/EllipticK[16*x], {x, 0, 20}], x]
%Y Cf. A010370, A054474, A261975, A328128.
%K sign
%O 0,2
%A _Vaclav Kotesovec_, Oct 04 2019
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