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%I #17 Dec 06 2022 09:40:04
%S 2,-4,10,-20,36,-64,110,-180,288,-452,692,-1044,1554,-2276,3296,-4724,
%T 6696,-9408,13108,-18112,24850,-33864,45844,-61696,82564,-109892,
%U 145536,-191828,251684,-328804,427802,-554408,715808,-920896,1180660,-1508736,1921896,-2440740,3090612
%N Expansion of the elliptic function sqrt(k(q))/q^(1/4) in powers of q, where sqrt(k(q)) = theta_2(q)/theta_3(q).
%D N. J. Fine, Basic Hypergeometric Series and Applications, Amer. Math. Soc., 1988; Eq. (34.3).
%F Expansion of 2 * q^(-1/4) * (eta(q) * eta(q^4)^2 / eta(q^2)^3)^2 in powers of q. - _Michael Somos_, Jun 12 2012
%F a(n) ~ (-1)^n * exp(Pi*sqrt(n))/(2^(5/2)*n^(3/4)). - _Vaclav Kotesovec_, Jul 04 2016
%F 2 * ({1} U (Euler transform of period 4 sequence [-2, 4, -2, 0])). - _Georg Fischer_, Dec 06 2022
%e 2 - 4*x + 10*x^2 - 20*x^3 + 36*x^4 - 64*x^5 + 110*x^6 - 180*x^7 + 288*x^8 - ...
%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) + ...
%t QP = QPochhammer; s = 2*(QP[q]*(QP[q^4]^2/QP[q^2]^3))^2 + O[q]^40; CoefficientList[s, q] (* _Jean-François Alcover_, Nov 27 2015, adapted from PARI *)
%t nmax = 50; CoefficientList[Series[2*Product[(1+x^(2*k))^4 / (1+x^k)^2, {k, 1, nmax}], {x, 0, nmax}], x] (* _Vaclav Kotesovec_, Jul 04 2016 *)
%o (PARI) {a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( 2 * (eta(x + A) * eta(x^4 + A)^2 / eta(x^2 + A)^3)^2, n))} /* _Michael Somos_, Jun 12 2012 */
%Y See A127391 for another version. Dividing by 2 gives A079006. Cf. A001936, A001938.
%K sign
%O 0,1
%A _N. J. A. Sloane_, Mar 31 2007
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