A038535 Numerators of coefficients of EllipticE/Pi.

1, -1, -3, -5, -175, -441, -4851, -14157, -2760615, -8690825, -112285459, -370263621, -19870814327, -67607800225, -931331941875, -3241035157725, -2913690606794775, -10313859829588425, -147068001273760875, -527570807893408125, -30451387031607516975

Offset: 0

Author

Wouter Meeussen, revised Jan 03 2001

Keywords

• frac
• sign

Comments

Contribution from Wolfdieter Lang, Nov 08 2010: (Start)

a(n)/A056982(n) = -(binomial(2*n,n)^2)/((2*n-1)*2^(4*n)), n>=0, are the coefficients of x^n of hypergeometric([1/2,-1/2],[1],x).

The series hypergeometric([1/2,-1/2],[1],e^2)=L/(2*Pi*a) with L the perimeter of an ellipse with major axis a and numerical eccentricity e. (End)

Crossrefs

Cf. A038533, A038534.

a(n) divides A000891(n+1).

Programs

Mathematica

Numerator[CoefficientList[Series[EllipticE[m]/Pi,{m,0,25}],m]] (* Harvey P. Dale, Dec 16 2011 *)

Formula

a(n) = 2^(-2 w[n])binomial[2n, n]^2 (-1)^(2n)/(1-2n) with w[n]=A000120 = number of 1's in binary expansion of n