login
A369538
Expansion of g.f. A(x) satisfying A(x) = 1 + 8*x * AGM(A(x), A(x)^2).
4
1, 8, 96, 1376, 21760, 366176, 6431488, 116551040, 2163118080, 40907835232, 785471061760, 15272052137856, 300077039734784, 5949171298710144, 118858435514103808, 2390669459946235392, 48369365721497534464, 983759515642369327456, 20101539919939043645184, 412461687626131640565632
OFFSET
0,2
COMMENTS
Here AGM(x,y) = AGM((x+y)/2, sqrt(x*y)) denotes the arithmetic-geometric mean.
LINKS
FORMULA
G.f. A(x) = Sum_{n>=0} a(n)*x^n satisfies the following formulas.
(1) A(x) = 1 + 8*x * AGM(A(x), A(x)^2).
(2) A(x) = 1 + 8*x * AGM(A(x)^(3/2), (A(x) + A(x)^2)/2).
(3) A(x) = 1 + 8 * Series_Reversion( x / AGM(1 + 8*x, (1 + 8*x)^2) ).
(4) A( x / AGM(1 + 8*x, (1 + 8*x)^2) ) = 1 + 8*x.
a(n) ~ c * d^n / n^(3/2), where d = 22.1630051344803196287731245642346070282303059361700001080950958441256... and c = 0.99743551254261758609104583646696482831141906954702821438454764216307... - Vaclav Kotesovec, Jan 29 2024
EXAMPLE
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 + ...
RELATED SERIES.
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 + ...
where
A( x/AGM(1 + 8*x, (1 + 8*x)^2) ) = 1 + 8*x.
MATHEMATICA
(* 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 *)
PROG
(PARI) /* From definition: A(x) = 1 + 8*x*AGM(A(x), A(x)^2) */
{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)}
for(n=0, 30, print1(a(n), ", "))
(PARI) /* From formula: A(x) = 1 + 8*x*AGM(A(x)^(3/2), (A(x) + A(x)^2)/2) */
{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)}
for(n=0, 30, print1(a(n), ", "))
(PARI) /* From A(x) = 1 + 8*Series_Reversion(x/AGM(1+8*x, (1+8*x)^2)) */
{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)}
for(n=0, 30, print1(a(n), ", "))
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Jan 28 2024
STATUS
approved