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A272823
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G.f. satisfies: A(x) = 1 + 4*x*AGM(A(x), A(x)^3).
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6
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1, 4, 32, 336, 4032, 52336, 715392, 10144192, 147836416, 2200709040, 33319564288, 511496462656, 7942988228608, 124551530359360, 1969386732874752, 31364967043386112, 502686338657607680, 8101474649157519536, 131212844750426749696, 2134554132316280052288, 34862936239396076532736, 571454628433830080180288, 9397626191659208856570880, 155006334509119865698297600, 2563700952357088703495372800
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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. satisfies:
(1) A(x) = 1 + 4*x*AGM(A(x)^2, (A(x) + A(x)^3)/2).
(2) A(x) = 1 + 4*Series_Reversion( x / AGM(1+4*x, (1+4*x)^3) ).
a(n) ~ c * d^n / n^(3/2), where d = 17.6088646774568498919315031912184045773663297219819943809841685080399155... and c = 0.3922200012562096239034743054558268956365939170567699740621520897631... - Vaclav Kotesovec, Nov 15 2023
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EXAMPLE
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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 +...
where A(x) = 1 + 4*x*AGM(A(x), A(x)^3).
RELATED SERIES.
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 +...
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 +...
(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 +...
where A(x) = 1 + 4*x*AGM(A(x)^2, (A(x) + A(x)^3)/2).
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 +...
where A(x) = 1 + 4*Series_Reversion( x / AGM(1+4*x, (1+4*x)^3)) ).
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MATHEMATICA
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(* 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 *)
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PROG
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(PARI) /* From definition: A(x) = 1 + 4*x*AGM(A(x), A(x)^3) */
{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)}
for(n=0, 30, print1(a(n), ", "))
(PARI) /* From formula: A(x) = 1 + 4*x*AGM(A(x)^2, (A(x) + A(x)^3)/2) */
{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)}
for(n=0, 30, print1(a(n), ", "))
(PARI) /* From A(x) = 1 + 4*Series_Reversion(x/AGM(1+4*x, (1+4*x)^3)) */
{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)}
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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