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

%I A127391 #19 Mar 02 2021 06:28:07
%S A127391 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 A127391 -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 A127391 -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 A127391 Series expansion of the elliptic function sqrt(k) = theta_2/theta_3 in powers of q^(1/4).
%C A127391 It appears that a(n) = 2 * A208933(n) - A212318(n) for n>0. - _Thomas Baruchel_, May 14 2018
%C A127391 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 A127391 N. J. Fine, Basic Hypergeometric Series and Applications, Amer. Math. Soc., 1988; Eq. (34.3).
%e A127391 2*x - 4*x^5 + 10*x^9 - 20*x^13 + 36*x^17 - 64*x^21 + 110*x^25 -180*x^29 + ...
%e A127391 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 A127391 See A127392 for another version. Dividing by 2 gives A079006. Cf. A001936, A001938.
%K A127391 sign
%O A127391 0,2
%A A127391 _N. J. A. Sloane_, Mar 31 2007