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A369539
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Expansion of g.f. A(x) satisfying A(x) = 1 + 8*x * AGM(A(x)^2, A(x)^3).
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4
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1, 8, 160, 4192, 125184, 4039264, 137183488, 4831873408, 174884458496, 6464875435872, 243049515606272, 9264347436276608, 357204831146577920, 13906950967902306944, 545951685104975276032, 21587442538147647608320, 858975581766808512823296, 34369283236381014527279456
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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 + 8*x * AGM(A(x)^2, A(x)^3).
(2) A(x) = 1 + 8*x * AGM(A(x)^(5/2), (A(x)^2 + A(x)^3)/2).
(3) A(x) = 1 + 8 * Series_Reversion( x / AGM((1 + 8*x)^2, (1 + 8*x)^3) ).
(4) A( x / AGM((1 + 8*x)^2, (1 + 8*x)^3) ) = 1 + 8*x.
a(n) ~ c * d^n / n^(3/2), where d = 43.7139872016060880921082193574226064477439580563964019841877818207326... and c = 0.32250297108028000960144303111184352981179935271075437927423118550208... - Vaclav Kotesovec, Jan 29 2024
A(1/d) = 1.6405711647668295617017794194853407... where d is given above. - Paul D. Hanna, Jan 29 2024
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EXAMPLE
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G.f.: A(x) = 1 + 8*x + 160*x^2 + 4192*x^3 + 125184*x^4 + 4039264*x^5 + 137183488*x^6 + 4831873408*x^7 + 174884458496*x^8 + 6464875435872*x^9 + 243049515606272*x^10 + ...
RELATED SERIES.
x/AGM((1 + 8*x)^2, (1 + 8*x)^3) = x - 20*x^2 + 276*x^3 - 3248*x^4 + 34980*x^5 - 356112*x^6 + 3487568*x^7 - 33204160*x^8 + 309415716*x^9 - 2835178320*x^10 + ...
where A( x/AGM((1 + 8*x)^2, (1 + 8*x)^3) ) = 1 + 8*x.
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MATHEMATICA
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(* Calculation of constants {d, c}: *) {1/r, s*(s - 1)*Sqrt[(1 + s)/(2*Pi*(4 + s - 7*s^2 + 4*s^3))]} /. FindRoot[{1 + 4*Pi*r*s^3/EllipticK[1 - 1/s^2] == s, 4*Pi*r*s*(2 + s - 2*s^2) + (-1 + s)*EllipticE[1 - 1/s^2] == 0}, {r, 1/50}, {s, 3/2}, WorkingPrecision -> 70] (* Vaclav Kotesovec, Jan 29 2024 *)
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PROG
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(PARI) /* From definition: A(x) = 1 + 8*x*AGM(A(x)^2, A(x)^3) */
{a(n) = my(A=1+4*x + x*O(x^n)); for(i=1, n, A = 1 + 8*x*agm(A^2, A^3)); polcoeff(A, n)}
for(n=0, 30, print1(a(n), ", "))
(PARI) /* From formula: A(x) = 1 + 8*x*AGM(A(x)^(5/2), (A(x)^2 + A(x)^3)/2) */
{a(n) = my(A=1+4*x + x*O(x^n)); for(i=1, n, A = 1 + 8*x*agm(A^(5/2), (A^2 + A^3)/2)); polcoeff(A, n)}
for(n=0, 30, print1(a(n), ", "))
(PARI) /* From A(x) = 1 + 8*Series_Reversion(x/AGM((1+8*x)^2, (1+8*x)^3)) */
{a(n) = my(A=1); A = 1 + 8*serreverse(x/agm((1+8*x)^2, (1+8*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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