A127391 Series expansion of the elliptic function sqrt(k) = theta_2/theta_3 in powers of q^(1/4).

0, 2, 0, 0, 0, -4, 0, 0, 0, 10, 0, 0, 0, -20, 0, 0, 0, 36, 0, 0, 0, -64, 0, 0, 0, 110, 0, 0, 0, -180, 0, 0, 0, 288, 0, 0, 0, -452, 0, 0, 0, 692, 0, 0, 0, -1044, 0, 0, 0, 1554, 0, 0, 0, -2276, 0, 0, 0, 3296, 0, 0, 0, -4724, 0, 0, 0, 6696, 0, 0, 0, -9408, 0, 0, 0, 13108, 0, 0, 0, -18112, 0, 0, 0, 24850, 0

Offset: 0

Author

N. J. A. Sloane, Mar 31 2007

Keywords

• sign

Comments

It appears that a(n) = 2 * A208933(n) - A212318(n) for n>0. - Thomas Baruchel, May 14 2018

Empirical: Sum_{n>=1} a(n)/exp(Pi*(n-1)) = 3 + 2*sqrt(2) - 2*sqrt(4 + 3*sqrt(2)). - Simon Plouffe, Mar 01 2021

Examples

			2*x - 4*x^5 + 10*x^9 - 20*x^13 + 36*x^17 - 64*x^21 + 110*x^25 -180*x^29 + ...
2*q^(1/4) - 4*q^(5/4) + 10*q^(9/4) - 20*q^(13/4) + 36*q^(17/4) - 64*q^(21/4) + ...
		

References

• N. J. Fine, Basic Hypergeometric Series and Applications, Amer. Math. Soc., 1988; Eq. (34.3).

Crossrefs

See A127392 for another version. Dividing by 2 gives A079006. Cf. A001936, A001938.