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%I #28 Jul 24 2017 08:24:15
%S 0,2,0,0,0,0,0,0,0,-6,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,10,0,0,0,0,0,0,0,
%T 0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-14,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,
%U 0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,18,0,0,0
%N Expansion of Jacobi theta function {theta_1}'(q) in powers of q^(1/4).
%D J. Tannery and J. Molk, Eléments de la Théorie des Fonctions Elliptiques, Vol. 2, Gauthier-Villars, Paris, 1902; Chelsea, NY, 1972, see p. 27.
%D E. T. Whittaker and G. N. Watson, A Course of Modern Analysis, Cambridge Univ. Press, 4th ed., 1963, p. 464.
%H Antti Karttunen, <a href="/A002483/b002483.txt">Table of n, a(n) for n = 0..10201</a>
%H J. H. Conway and N. J. A. Sloane, <a href="http://dx.doi.org/10.1007/978-1-4757-2016-7">Sphere Packings, Lattices and Groups</a>, Springer-Verlag, p. 102.
%F Expansion of 2 * q^(-1/8) * eta(q)^3 in powers of q. - _Michael Somos_, May 31 2012
%F G.f.: 2 * x * Product_{k>0} (1 - x^(8*k))^3. - _Michael Somos_, May 31 2012
%F For n > 0, a(n) = (((1/8)*(4*t^2 + 4*t + 1 - n) - 1)*4 + 2)*(t-r)*(-1)^(t+1), where t = floor((sqrt(n)+1)/2) and r = floor((sqrt(n-1)+1)/2). - _Mikael Aaltonen_, Jan 16 2015
%e 2*x - 6*x^9 + 10*x^25 - 14*x^49 + 18*x^81 - 22*x^121 + 26*x^169 - 30*x^225 + ...
%p Sum( (-1)^m*(2*m+1)*q^ ( ((2*m+1)/2)^2 ), m=-10,10);
%t a[ n_] := SeriesCoefficient[ EllipticThetaPrime[ 1, 0, q], {q, 0, n/4}] (* _Michael Somos_, May 31 2012 *)
%t s = 2q*QPochhammer[q^8]^3+O[q]^90; CoefficientList[s, q] (* _Jean-François Alcover_, Nov 30 2015, adapted from PARI *)
%o (PARI) {a(n) = local(m); if( issquare( n, &m) && m%2, 2 * (-1)^(m \ 2) * m, 0)} /* _Michael Somos_, May 31 2012 */
%o (PARI) {a(n) = if( n<1, 0, n--; polcoeff( 2 * eta(x^8 + x * O(x^n))^3, n))} /* _Michael Somos_, May 31 2012 */
%Y Dividing by 2 gives (essentially) A245552.
%Y See also the g.f. for A053187.
%K sign
%O 0,2
%A _N. J. A. Sloane_
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