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%I #7 Oct 03 2019 09:53:50
%S 1,8,56,384,2648,18496,131008,940032,6821848,49985984,369258560,
%T 2746629120,20549693888,154518118912,1166873394688,8844937101312,
%U 67265481552856,513038965707968,3923108472072512,30068733313938432,230943237733355840,1777114026405752320
%N Expansion of 1 / AGM(1, 1 - 8*x)^2 in powers of x.
%C AGM(x,y) = AGM((x+y)/2,sqrt(x*y)) is the arithmetic-geometric mean.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/Arithmetic-GeometricMean.html">Arithmetic-Geometric Mean</a>
%F Recurrence: n^3*a(n) = 4*(2*n - 1)*(3*n^2 - 3*n + 2)*a(n-1) - 16*(n-1)*(13*n^2 - 26*n + 20)*a(n-2) + 128*(2*n - 3)*(3*n^2 - 9*n + 8)*a(n-3) - 1024*(n-2)^3*a(n-4).
%F a(n) ~ 2^(3*n + 3) * (log(4*n) + gamma) / (Pi^2 * n), where gamma is the Euler-Mascheroni constant A001620.
%t CoefficientList[Series[(2*EllipticK[1/(1 - 1/(4*x))^2]/(Pi*(1 - 4*x)))^2, {x, 0, 25}], x]
%t CoefficientList[Series[Hypergeometric2F1[1/2, 1/2, 1, 16*x*(1 - 4*x)]^2, {x, 0, 25}], x]
%Y Cf. A081085, A089603.
%K nonn
%O 0,2
%A _Vaclav Kotesovec_, Sep 27 2019
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