A171454 G.f. satisfies: A(x) = 1 + 4*x*AGM(1, A(x)^2).

1, 4, 16, 80, 448, 2672, 16640, 106944, 704000, 4722608, 32166784, 221865280, 1546491904, 10876777024, 77091573760, 550088739584, 3948410757120, 28489277352112, 206520803651712, 1503353875355200, 10984898330047488, 80540719266134080, 592362120108263424, 4369140213882013440

Offset

0,2

Links

Vaclav Kotesovec, Table of n, a(n) for n = 0..400

Formula

a(n) ~ c * d^n / n^(3/2), where d = 7.862810359254200633908490328120234046255594283562932892563... and c = 1.2926544621133576475917023125188188972684483736846308027... - Vaclav Kotesovec, Nov 15 2023

Example

G.f.: A(x) = 1 + 4*x + 16*x^2 + 80*x^3 + 448*x^4 + 2672*x^5 + ...
A(x)^2 = 1 + 8*x + 48*x^2 + 288*x^3 + 1792*x^4 + 11488*x^5 + ...
AGM(1, A(x)^2) = 1 + 4*x + 20*x^2 + 112*x^3 + 668*x^4 + 4160*x^5 + ...

Mathematica

(* Calculation of constants {d, c}: *) {1/r, Sqrt[r*s*(-1 - s + s^4 + s^5) / (2*Pi*r*(-1 - 4*s^2 + s^4) + 8*s*EllipticK[(-1 + s^2)^2/(1 + s^2)^2])]} /. FindRoot[{(s - 1)/(4*r) == Pi*s^2/(2*EllipticK[1 - 1/s^4]), EllipticE[(-1 + s^2)^2/(1 + s^2)^2] == Pi*r*s}, {r, 1/8}, {s, 3}, WorkingPrecision -> 70] (* Vaclav Kotesovec, Nov 15 2023 *)

Prog

(PARI) {a(n)=local(A=1+x+x*O(x^n)); for(i=1, n, A=1+4*x*agm(1, A^2)); polcoeff(A, n)}

Crossrefs

Cf. A060691.

Sequence in context: A183146 A160564 A075581 * A316944 A020080 A279361

Adjacent sequences: A171451 A171452 A171453 * A171455 A171456 A171457

Keyword

nonn

Author

Paul D. Hanna, Dec 09 2009

Extensions

More terms from Jinyuan Wang, Feb 25 2020

Status

approved