OFFSET
0,2
COMMENTS
Here AGM(x,y) = AGM((x+y)/2,sqrt(x*y)) is the arithmetic-geometric mean.
LINKS
Vaclav Kotesovec, Table of n, a(n) for n = 0..650
FORMULA
Let B(x) = Sum_{n>=0} C(2*n,n)^2 * x^n, then g.f. A(x) satisfies:
(1) A(x) = B(x*A(x)).
(2) B(x) = A(x/B(x)).
(3) A(x) = 1/x * Series_Reversion(x/B(x)).
a(n) ~ c * d^n / n^(3/2), where d = 28.60714633732041935517418963604715019406455947751016145247296001862... and c = 0.25701137473154058587219119456391566506312220485172145290097621... - Vaclav Kotesovec, Nov 15 2023
EXAMPLE
G.f.: A(x) = 1 + 4*x + 52*x^2 + 896*x^3 + 17604*x^4 + 373408*x^5 +...
such that
A(x) = 1 + 4*x*A(x) + 36*x^2*A(x)^2 + 400*x^3*A(x)^3 + 4900*x^4*A(x)^4 + 63504*x^5*A(x)^5 +...+ C(2*n,n)^2 * x^n * A(x)^n +...
MATHEMATICA
(* Calculation of constants {d, c}: *) {1/r, s*Sqrt[(1 - 16*r*s)/(2*Pi*(36*r*s - 1))]} /. FindRoot[{Pi*s*Sqrt[1 - 16*r*s] == 2*EllipticK[16*r*s/(-1 + 16*r*s)], 3*Pi*s*Sqrt[1 - 16*r*s] == 2*EllipticE[1 + 1/(-1 + 16*r*s)]}, {r, 1/28}, {s, 3/2}, WorkingPrecision -> 70] (* Vaclav Kotesovec, Nov 15 2023 *)
PROG
(PARI) {a(n)=local(A=1); for(i=1, n, A = 1 / agm(1, sqrt(1 - 16*x*A +x*O(x^n)))); polcoeff(A, n)}
for(n=0, 20, print1(a(n), ", "))
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Sep 09 2014
STATUS
approved