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%I #13 Jan 29 2024 04:46:15
%S 1,4,32,384,5376,81920,1318912,22071296,380084224,6691479552,
%T 119890509824,2178958163968,40073602269184,744399420391424,
%U 13946358907011072,263220821247393792,5000085343337185280,95520905055747178496,1834027221478623150080,35372549509799248658432
%N Expansion of g.f. A(x) satisfying A(x) = 1 + 4*x * AGM(1, A(x)^4).
%C Here AGM(x,y) = AGM((x+y)/2, sqrt(x*y)) denotes the arithmetic-geometric mean.
%H Paul D. Hanna, <a href="/A369536/b369536.txt">Table of n, a(n) for n = 0..400</a>
%F G.f. A(x) = Sum_{n>=0} a(n)*x^n satisfies the following formulas.
%F (1) A(x) = 1 + 4*x * AGM(1, A(x)^4).
%F (2) A(x) = 1 + 4*x * AGM(A(x)^2, (1 + A(x)^4)/2).
%F (3) A(x) = 1 + 4 * Series_Reversion( x / AGM(1, (1 + 4*x)^4) ).
%F (4) A( x/AGM(1, (1 + 4*x)^4) ) = 1 + 4*x.
%F a(n) ~ c * d^n / n^(3/2), where d = 20.8911293747878758394214491571395886690885608604807120892771607914028... and c = 0.2494539611266913248489641272521896595054291412784920760863145867198... - _Vaclav Kotesovec_, Jan 29 2024
%e G.f.: A(x) = 1 + 4*x + 32*x^2 + 384*x^3 + 5376*x^4 + 81920*x^5 + 1318912*x^6 + 22071296*x^7 + 380084224*x^8 + 6691479552*x^9 + 119890509824*x^10 + ...
%e RELATED SERIES.
%e x/AGM(1, (1 + 4*x)^4) = x - 8*x^2 + 32*x^3 - 64*x^4 + 3584*x^7 - 22528*x^8 + 34816*x^9 + 245760*x^10 - 1163264*x^11 - 3211264*x^12 + ...
%e where A( x/AGM(1, (1 + 4*x)^4) ) = 1 + 4*x.
%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[{1 + 2*Pi*r*s^4/EllipticK[1 - 1/s^8] == s, (s^8 - 1)/(s - 1) + 2*(s - 1)*s^3 * EllipticE[1 - 1/s^8]/(Pi*r) == 4*s^7}, {r, 1/20}, {s, 2}, WorkingPrecision -> 70] (* _Vaclav Kotesovec_, Jan 29 2024 *)
%o (PARI) /* From definition: A(x) = 1 + 4*x*AGM(1, A(x)^4) */
%o {a(n)=my(A=1+4*x + x*O(x^n)); for(i=1, n, A = 1 + 4*x*agm(1, A^4)); polcoeff(A, n)}
%o for(n=0, 30, print1(a(n), ", "))
%o (PARI) /* From formula: A(x) = 1 + 4*x*AGM(A(x)^2, (1 + A(x)^4)/2) */
%o {a(n)=my(A=1+4*x + x*O(x^n)); for(i=1, n, A = 1 + 4*x*agm(A^2, (1 + A^4)/2)); polcoeff(A, n)}
%o for(n=0, 30, print1(a(n), ", "))
%o (PARI) /* From A(x) = 1 + 4*Series_Reversion(x/AGM(1, (1+4*x)^4)) */
%o {a(n) = my(A=1); A = 1 + 4*serreverse(x/agm(1, (1+4*x)^4 +x*O(x^n))); polcoeff(A, n)}
%o for(n=0, 30, print1(a(n), ", "))
%Y Cf. A171454, A272823, A369537, A369538, A369539.
%K nonn
%O 0,2
%A _Paul D. Hanna_, Jan 28 2024
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