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A369536
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Expansion of g.f. A(x) satisfying A(x) = 1 + 4*x * AGM(1, A(x)^4).
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4
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1, 4, 32, 384, 5376, 81920, 1318912, 22071296, 380084224, 6691479552, 119890509824, 2178958163968, 40073602269184, 744399420391424, 13946358907011072, 263220821247393792, 5000085343337185280, 95520905055747178496, 1834027221478623150080, 35372549509799248658432
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OFFSET
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0,2
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COMMENTS
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Here AGM(x,y) = AGM((x+y)/2, sqrt(x*y)) denotes the arithmetic-geometric mean.
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LINKS
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FORMULA
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G.f. A(x) = Sum_{n>=0} a(n)*x^n satisfies the following formulas.
(1) A(x) = 1 + 4*x * AGM(1, A(x)^4).
(2) A(x) = 1 + 4*x * AGM(A(x)^2, (1 + A(x)^4)/2).
(3) A(x) = 1 + 4 * Series_Reversion( x / AGM(1, (1 + 4*x)^4) ).
(4) A( x/AGM(1, (1 + 4*x)^4) ) = 1 + 4*x.
a(n) ~ c * d^n / n^(3/2), where d = 20.8911293747878758394214491571395886690885608604807120892771607914028... and c = 0.2494539611266913248489641272521896595054291412784920760863145867198... - Vaclav Kotesovec, Jan 29 2024
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EXAMPLE
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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 + ...
RELATED SERIES.
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 + ...
where A( x/AGM(1, (1 + 4*x)^4) ) = 1 + 4*x.
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MATHEMATICA
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(* 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 *)
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PROG
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(PARI) /* From definition: A(x) = 1 + 4*x*AGM(1, A(x)^4) */
{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)}
for(n=0, 30, print1(a(n), ", "))
(PARI) /* From formula: A(x) = 1 + 4*x*AGM(A(x)^2, (1 + A(x)^4)/2) */
{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)}
for(n=0, 30, print1(a(n), ", "))
(PARI) /* From A(x) = 1 + 4*Series_Reversion(x/AGM(1, (1+4*x)^4)) */
{a(n) = my(A=1); A = 1 + 4*serreverse(x/agm(1, (1+4*x)^4 +x*O(x^n))); polcoeff(A, n)}
for(n=0, 30, print1(a(n), ", "))
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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