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%I #7 Jan 01 2024 14:08:14
%S 1,27,2250,385875,112521150,49921883550,31336679474100,
%T 26440323306271875,28866957423047493750,39599692192936551926250,
%U 66678681708870074070727500,135213253391970365203090248750
%N 1/2+Sum_{n >= 1} a(n)*x^(2*n)/(4^n*(2*n)!) = 1/Pi*EllipticK(x).
%F From _Vaclav Kotesovec_, Nov 14 2023: (Start)
%F a(n) = binomial(2*n,n) * binomial(2*n-1,n) * (2*n)! / 4^n.
%F a(n) ~ 2^(4*n) * n^(2*n - 1/2) / (sqrt(Pi) * exp(2*n)). (End)
%e EllipticK(x) = 1/2*Pi + 1/8*Pi*x^2 + 9/128*Pi*x^4 + 25/512*Pi*x^6 + 1225/32768*Pi*x^8 + 3969/131072*Pi*x^10 + O(x^12).
%t Table[Binomial[2*n, n]*Binomial[2*n - 1, n]*(2*n)!/4^n, {n, 1, 20}] (* _Vaclav Kotesovec_, Nov 14 2023 *)
%Y Cf. A038533, A038534.
%K nonn
%O 1,2
%A _Vladeta Jovovic_, Apr 14 2001
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