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

Author

Michael Somos, Feb 16 2004

Keywords

• nonn

Programs

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 *)

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

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