A171455 G.f. satisfies: A(x) = 1 + 2*x*AGM(1, A(x)^4).

1, 2, 8, 48, 336, 2560, 20608, 172432, 1484704, 13069296, 117080576, 1063944416, 9783594304, 90869069872, 851218195008, 8032861976544, 76295247548480, 728766670652368, 6996258626856320, 67467783946608064, 653254749175955584, 6348266152788407648, 61896814517299122560

Offset

0,2

Links

Vaclav Kotesovec, Table of n, a(n) for n = 0..235

Formula

a(n) ~ c * d^n / n^(3/2), where d = 10.4455646873939379197107245785697943345442804302403560446385803957... and c = 0.249453961126691324848964127252189659505429141278492076086314586719... - Vaclav Kotesovec, Nov 15 2023

Example

G.f.: A(x) = 1 + 2*x + 8*x^2 + 48*x^3 + 336*x^4 + 2560*x^5 + ...
A(x)^2 = 1 + 4*x + 20*x^2 + 128*x^3 + 928*x^4 + 7232*x^5 + ...
A(x)^4 = 1 + 8*x + 56*x^2 + 416*x^3 + 3280*x^4 + 27008*x^5 + ...
AGM(1, A(x)^4) = 1 + 4*x + 24*x^2 + 168*x^3 + 1280*x^4 + 10304*x^5 + ...

Mathematica

(* Calculation of constants {d, c}: *) {1/r, Sqrt[s*(1 - s - s^8 + s^9) / (2*Pi*(1 + 2*s + 2*s^2 + 2*s^3 + 2*s^4 + 2*s^5 + 2*s^6 - 14*s^7 + 9*s^8))]} /. FindRoot[{Pi*r*s^4 / EllipticK[1 - 1/s^8] == s - 1, -4*Pi*r*s^7 + Pi*r*(-1 + s^8)/(-1 + s) + 4*(-1 + s)*s^3 * EllipticE[1 - 1/s^8] == 0}, {r, 1/10}, {s, 3/2}, 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+2*x*agm(1, A^4)); polcoeff(A, n)}

Crossrefs

Cf. A171454.

Sequence in context: A007170 A355488 A379382 * A136722 A085615 A054726

Adjacent sequences: A171452 A171453 A171454 * A171456 A171457 A171458

Keyword

nonn

Author

Paul D. Hanna, Dec 09 2009

Extensions

More terms from Jinyuan Wang, Feb 25 2020

Status

approved