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A369538 Expansion of g.f. A(x) satisfying A(x) = 1 + 8*x * AGM(A(x), A(x)^2).

%I A369538 #9 Jan 29 2024 05:40:52
%S A369538 1,8,96,1376,21760,366176,6431488,116551040,2163118080,40907835232,
%T A369538 785471061760,15272052137856,300077039734784,5949171298710144,
%U A369538 118858435514103808,2390669459946235392,48369365721497534464,983759515642369327456,20101539919939043645184,412461687626131640565632
%N A369538 Expansion of g.f. A(x) satisfying A(x) = 1 + 8*x * AGM(A(x), A(x)^2).
%C A369538 Here AGM(x,y) = AGM((x+y)/2, sqrt(x*y)) denotes the arithmetic-geometric mean.
%H A369538 Paul D. Hanna, <a href="/A369538/b369538.txt">Table of n, a(n) for n = 0..400</a>
%F A369538 G.f. A(x) = Sum_{n>=0} a(n)*x^n satisfies the following formulas.
%F A369538 (1) A(x) = 1 + 8*x * AGM(A(x), A(x)^2).
%F A369538 (2) A(x) = 1 + 8*x * AGM(A(x)^(3/2), (A(x) + A(x)^2)/2).
%F A369538 (3) A(x) = 1 + 8 * Series_Reversion( x / AGM(1 + 8*x, (1 + 8*x)^2) ).
%F A369538 (4) A( x / AGM(1 + 8*x, (1 + 8*x)^2) ) = 1 + 8*x.
%F A369538 a(n) ~ c * d^n / n^(3/2), where d = 22.1630051344803196287731245642346070282303059361700001080950958441256... and c = 0.99743551254261758609104583646696482831141906954702821438454764216307... - _Vaclav Kotesovec_, Jan 29 2024
%e A369538 G.f.: A(x) = 1 + 8*x + 96*x^2 + 1376*x^3 + 21760*x^4 + 366176*x^5 + 6431488*x^6 + 116551040*x^7 + 2163118080*x^8 + 40907835232*x^9 + 785471061760*x^10 + ...
%e A369538 RELATED SERIES.
%e A369538 x / AGM(1 + 8*x, (1 + 8*x)^2) = x - 12*x^2 + 116*x^3 - 1040*x^4 + 8996*x^5 - 76272*x^6 + 638672*x^7 - 5303616*x^8 + 43782436*x^9 - 359852592*x^10 + ...
%e A369538 where
%e A369538 A( x/AGM(1 + 8*x, (1 + 8*x)^2) ) = 1 + 8*x.
%t A369538 (* Calculation of constants {d,c}: *) {1/r, s*(s - 1)*Sqrt[(1 + s)/(2*Pi*(1 + 2*s - 2*s^2 + s^3))]} /. FindRoot[{1 + 4*Pi*r*s^2/EllipticK[1 - 1/s^2] == s, 4*Pi*r*(-1 - s + s^2) == (s - 1)*EllipticE[1 - 1/s^2]}, {r, 1/25}, {s, 5/2}, WorkingPrecision -> 70] (* _Vaclav Kotesovec_, Jan 29 2024 *)
%o A369538 (PARI) /* From definition: A(x) = 1 + 8*x*AGM(A(x), A(x)^2) */
%o A369538 {a(n) = my(A=1+4*x + x*O(x^n)); for(i=1, n, A = 1 + 8*x*agm(A, A^2)); polcoeff(A, n)}
%o A369538 for(n=0, 30, print1(a(n), ", "))
%o A369538 (PARI) /* From formula: A(x) = 1 + 8*x*AGM(A(x)^(3/2), (A(x) + A(x)^2)/2) */
%o A369538 {a(n) = my(A=1+4*x + x*O(x^n)); for(i=1, n, A = 1 + 8*x*agm(A^(3/2), (A + A^2)/2)); polcoeff(A, n)}
%o A369538 for(n=0, 30, print1(a(n), ", "))
%o A369538 (PARI) /* From A(x) = 1 + 8*Series_Reversion(x/AGM(1+8*x, (1+8*x)^2)) */
%o A369538 {a(n) = my(A=1); A = 1 + 8*serreverse(x/agm(1+8*x, (1+8*x)^2 +x*O(x^n))); polcoeff(A, n)}
%o A369538 for(n=0, 30, print1(a(n), ", "))
%Y A369538 Cf. A171454, A272823, A369536, A369537, A369539.
%K A369538 nonn
%O A369538 0,2
%A A369538 _Paul D. Hanna_, Jan 28 2024