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A188267 Coefficient of x^n in the series 1/(1-x*F(1/2,1/2;1;16x)), where F(a1,a2;b;x) is the hypergeometric series.

%I A188267 #28 Oct 03 2019 10:58:48
%S A188267 1,1,5,45,501,6161,80189,1082649,14996021,211674805,3031568597,
%T A188267 43920006709,642265758053,9465144429045,140400306506101,
%U A188267 2094220410467877,31387767877371013,472406259202624889,7136241394473619133,108153547914919084017
%N A188267 Coefficient of x^n in the series 1/(1-x*F(1/2,1/2;1;16x)), where F(a1,a2;b;x) is the hypergeometric series.
%C A188267 Equivalently, coefficient of x^n in the series 1/(1-(2x/Pi)*K(16x)), where K(x) = (Pi/2)*F(1/2,1/2;1;x) is the complete elliptic integral (defined as in Mathematica, i.e. with x instead of x^2).
%H A188267 Vincenzo Librandi, <a href="/A188267/b188267.txt">Table of n, a(n) for n = 0..800</a>
%H A188267 Vaclav Kotesovec, <a href="/A188267/a188267.jpg">Graph - the asymptotic ratio (10000 terms)</a>
%F A188267 Recurrence: a(n+1) = sum(binomial(2k,k)^2*a(n-k),k=0..n).
%F A188267 G.f.: 1/(1 - x/AGM(sqrt(1 - 16*x), 1)). - _Vaclav Kotesovec_, Sep 30 2019
%F A188267 a(n) ~ Pi * 2^(4*n + 4) / (n * (log(n) - 16*Pi)^2) * (1 - 2*(gamma + 4*log(2)) / (log(n) - 16*Pi) + (3*gamma^2 - Pi^2/2 + 24*gamma*log(2) + 48*log(2)^2) / (log(n) - 16*Pi)^2), where gamma is the Euler-Mascheroni constant A001620. - _Vaclav Kotesovec_, Oct 01 2019
%t A188267 CoefficientList[Series[1/(1-(2x/Pi)EllipticK[16x]),{x,0,100}],x]
%t A188267 a[0] = 1; Flatten[{1, Table[a[n+1] = Sum[Binomial[2*k, k]^2*a[n-k], {k, 0, n}], {n, 0, 20}]}] (* _Vaclav Kotesovec_, Sep 28 2019 *)
%Y A188267 Cf. A188266, A328046.
%K A188267 nonn,easy
%O A188267 0,3
%A A188267 _Emanuele Munarini_, Mar 30 2011