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G.f. satisfies: A(x) = 1 + 2*x*AGM(1, A(x)^4).
1

%I #13 Nov 15 2023 09:13:04

%S 1,2,8,48,336,2560,20608,172432,1484704,13069296,117080576,1063944416,

%T 9783594304,90869069872,851218195008,8032861976544,76295247548480,

%U 728766670652368,6996258626856320,67467783946608064,653254749175955584,6348266152788407648,61896814517299122560

%N G.f. satisfies: A(x) = 1 + 2*x*AGM(1, A(x)^4).

%H Vaclav Kotesovec, <a href="/A171455/b171455.txt">Table of n, a(n) for n = 0..235</a>

%F a(n) ~ c * d^n / n^(3/2), where d = 10.4455646873939379197107245785697943345442804302403560446385803957... and c = 0.249453961126691324848964127252189659505429141278492076086314586719... - _Vaclav Kotesovec_, Nov 15 2023

%e G.f.: A(x) = 1 + 2*x + 8*x^2 + 48*x^3 + 336*x^4 + 2560*x^5 + ...

%e A(x)^2 = 1 + 4*x + 20*x^2 + 128*x^3 + 928*x^4 + 7232*x^5 + ...

%e A(x)^4 = 1 + 8*x + 56*x^2 + 416*x^3 + 3280*x^4 + 27008*x^5 + ...

%e AGM(1, A(x)^4) = 1 + 4*x + 24*x^2 + 168*x^3 + 1280*x^4 + 10304*x^5 + ...

%t (* Calculation of constants {d,c}: *) {1/r, Sqrt[s*(1 - s - s^8 + s^9) / (2*Pi*(1 + 2*s + 2*s^2 + 2*s^3 + 2*s^4 + 2*s^5 + 2*s^6 - 14*s^7 + 9*s^8))]} /. FindRoot[{Pi*r*s^4 / EllipticK[1 - 1/s^8] == s - 1, -4*Pi*r*s^7 + Pi*r*(-1 + s^8)/(-1 + s) + 4*(-1 + s)*s^3 * EllipticE[1 - 1/s^8] == 0}, {r, 1/10}, {s, 3/2}, WorkingPrecision -> 70] (* _Vaclav Kotesovec_, Nov 15 2023 *)

%o (PARI) {a(n)=local(A=1+x+x*O(x^n));for(i=1,n,A=1+2*x*agm(1,A^4));polcoeff(A,n)}

%Y Cf. A171454.

%K nonn

%O 0,2

%A _Paul D. Hanna_, Dec 09 2009

%E More terms from _Jinyuan Wang_, Feb 25 2020