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%I #12 Nov 15 2023 08:23:20
%S 1,4,32,336,4032,52336,715392,10144192,147836416,2200709040,
%T 33319564288,511496462656,7942988228608,124551530359360,
%U 1969386732874752,31364967043386112,502686338657607680,8101474649157519536,131212844750426749696,2134554132316280052288,34862936239396076532736,571454628433830080180288,9397626191659208856570880,155006334509119865698297600,2563700952357088703495372800
%N G.f. satisfies: A(x) = 1 + 4*x*AGM(A(x), A(x)^3).
%C Here AGM(x,y) = AGM((x+y)/2, sqrt(x*y)) denotes the arithmetic-geometric mean.
%H Vaclav Kotesovec, <a href="/A272823/b272823.txt">Table of n, a(n) for n = 0..630</a>
%F G.f. satisfies:
%F (1) A(x) = 1 + 4*x*AGM(A(x)^2, (A(x) + A(x)^3)/2).
%F (2) A(x) = 1 + 4*Series_Reversion( x / AGM(1+4*x, (1+4*x)^3) ).
%F a(n) ~ c * d^n / n^(3/2), where d = 17.6088646774568498919315031912184045773663297219819943809841685080399155... and c = 0.3922200012562096239034743054558268956365939170567699740621520897631... - _Vaclav Kotesovec_, Nov 15 2023
%e G.f.: A(x) = 1 + 4*x + 32*x^2 + 336*x^3 + 4032*x^4 + 52336*x^5 + 715392*x^6 + 10144192*x^7 + 147836416*x^8 + 2200709040*x^9 + 33319564288*x^10 +...
%e where A(x) = 1 + 4*x*AGM(A(x), A(x)^3).
%e RELATED SERIES.
%e A(x)^2 = 1 + 8*x + 80*x^2 + 928*x^3 + 11776*x^4 + 158432*x^5 + 2220416*x^6 + 32070528*x^7 + 474038272*x^8 + 7136118624*x^9 + 109031206528*x^10 +...
%e A(x)^3 = 1 + 12*x + 144*x^2 + 1840*x^3 + 24768*x^4 + 346704*x^5 + 4999424*x^6 + 73774656*x^7 + 1108876800*x^8 + 16918514448*x^9 + 261355433856*x^10 +...
%e (A(x) + A(x)^3)/2 = 1 + 8*x + 88*x^2 + 1088*x^3 + 14400*x^4 + 199520*x^5 + 2857408*x^6 + 41959424*x^7 + 628356608*x^8 + 9559611744*x^9 + 147337499072*x^10 +...
%e where A(x) = 1 + 4*x*AGM(A(x)^2, (A(x) + A(x)^3)/2).
%e AGM(1+4*x, (1+4*x)^3)) = 1 + 8*x + 20*x^2 + 16*x^3 - 4*x^4 + 16*x^6 - 64*x^7 + 172*x^8 - 352*x^9 + 560*x^10 - 832*x^11 + 2512*x^12 - 13568*x^13 + 65984*x^14 +...
%e where A(x) = 1 + 4*Series_Reversion( x / AGM(1+4*x, (1+4*x)^3)) ).
%t (* Calculation of constants {d,c}: *) {1/r, s*(s - 1)*Sqrt[(1 + s)*(1 + s^2) / (2*Pi*(1 + 2*s + 2*s^2 + 2*s^3 - 7*s^4 + 4*s^5))]} /. FindRoot[{2*Pi*r*s^3 / EllipticK[1 - 1/s^4] == s - 1, Pi*r*(1 + s + s^2 + s^3 - 2*s^4) == (1 - s)*s * EllipticE[1 - 1/s^4]}, {r, 1/18}, {s, 3/2}, WorkingPrecision -> 80] (* _Vaclav Kotesovec_, Nov 15 2023 *)
%o (PARI) /* From definition: A(x) = 1 + 4*x*AGM(A(x), A(x)^3) */
%o {a(n)=local(A=1+4*x + x*O(x^n)); for(i=1, n, A = 1 + 4*x*agm(A, A^3)); polcoeff(A, n)}
%o for(n=0,30,print1(a(n),", "))
%o (PARI) /* From formula: A(x) = 1 + 4*x*AGM(A(x)^2, (A(x) + A(x)^3)/2) */
%o {a(n)=local(A=1+4*x + x*O(x^n)); for(i=1, n, A = 1 + 4*x*agm(A^2, (A + A^3)/2)); polcoeff(A, n)}
%o for(n=0,30,print1(a(n),", "))
%o (PARI) /* From A(x) = 1 + 4*Series_Reversion(x/AGM(1+4*x, (1+4*x)^3)) */
%o {a(n) = my(A=1); A = 1 + 4*serreverse(x/agm(1+4*x,(1+4*x)^3 +x*O(x^n))); polcoeff(A,n)}
%o for(n=0,30,print1(a(n),", "))
%Y Cf. A171454, A272822.
%K nonn
%O 0,2
%A _Paul D. Hanna_, May 07 2016
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