A092266 Expansion of (1+4x)/AGM(1+4x,1-4x) where AGM denotes the arithmetic-geometric mean.

1, 4, 4, 16, 36, 144, 400, 1600, 4900, 19600, 63504, 254016, 853776, 3415104, 11778624, 47114496, 165636900, 662547600, 2363904400, 9455617600, 34134779536, 136539118144, 497634306624, 1990537226496, 7312459672336

Offset

0,2

Links

Table of n, a(n) for n=0..24.

Formula

G.f.: (1+4x)/AGM(1+4x, 1-4x) where AGM(x, y) is the arithmetic-geometric mean of Gauss and Legendre.

a(n) = A063886(n)^2.

a(2n) = A002894(n); a(2n+1) = 4*a(2n).

a(n) ~ 2^(2*n + 1) / (Pi*n). - Vaclav Kotesovec, Sep 27 2019

Mathematica

CoefficientList[Series[2*(1 + 4*x)*EllipticK[1 - (1 + 4*x)^2/(1 - 4*x)^2] / (Pi*(1 - 4*x)), {x, 0, 30}], x] (* Vaclav Kotesovec, Sep 27 2019 *)

Prog

(PARI) a(n)=((n==0)+2*binomial(n-1, (n-1)\2))^2;
(PARI) Vec( 1/agm(1, (1-4*x)/(1+4*x)+O(x^66)) ) \\ Joerg Arndt, Aug 14 2013

Crossrefs

Sequence in context: A038234 A099462 A218051 * A257606 A219398 A222104

Adjacent sequences: A092263 A092264 A092265 * A092267 A092268 A092269

Keyword

nonn

Author

Michael Somos, Feb 16 2004

Status

approved