%I #9 Nov 16 2023 02:25:00
%S 1,8,40,176,752,3168,13200,54592,224624,920672,3762240,15337408,
%T 62404656,253505184,1028426432,4167385088,16870487440,68237238816,
%U 275798010400,1113973454464,4496809953248,18142920856576,73165420555648
%N G.f. satisfies: AGM(1,A(x)) = 1/(1 - 4*x).
%H Vaclav Kotesovec, <a href="/A171189/b171189.txt">Table of n, a(n) for n = 0..1000</a>
%e G.f.: A(x) = 1 + 8*x + 40*x^2 + 176*x^3 + 752*x^4 + 3168*x^5 +...
%e A(x)^(1/2) = 1 + 4*x + 12*x^2 + 40*x^3 + 144*x^4 + 528*x^5 + 1960*x^6 +...
%e (1+A(x))/2 = 1 + 4*x + 20*x^2 + 88*x^3 + 376*x^4 + 1584*x^5 + 6600*x^6 +...
%t CoefficientList[InverseSeries[Series[1/4 - EllipticK[1 - 1/x^2]/(2*Pi*x), {x, 1, 25}], x], x] (* _Vaclav Kotesovec_, Nov 15 2023 *)
%o (PARI) {a(n)=local(A=1+8*x+sum(m=2,n-1,a(m)*x^m)+x^2*O(x^n));if(n<2,polcoeff(A,n),2*4^n-2*polcoeff(agm(1,A),n))}
%Y Cf. A171188.
%K nonn
%O 0,2
%A _Paul D. Hanna_, Dec 11 2009