A127391 - OEIS

%I #19 Mar 02 2021 06:28:07

%S 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,

%T -180,0,0,0,288,0,0,0,-452,0,0,0,692,0,0,0,-1044,0,0,0,1554,0,0,0,

%U -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

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

%C It appears that a(n) = 2 * A208933(n) - A212318(n) for n>0. - _Thomas Baruchel_, May 14 2018

%C 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

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

%e 2*x - 4*x^5 + 10*x^9 - 20*x^13 + 36*x^17 - 64*x^21 + 110*x^25 -180*x^29 + ...

%e 2*q^(1/4) - 4*q^(5/4) + 10*q^(9/4) - 20*q^(13/4) + 36*q^(17/4) - 64*q^(21/4) + ...

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

%K sign

%O 0,2

%A _N. J. A. Sloane_, Mar 31 2007