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%I #31 Mar 12 2018 10:13:47
%S 1,-4,12,-32,76,-168,352,-704,1356,-2532,4600,-8160,14176,-24168,
%T 40512,-66880,108876,-174984,277932,-436640,679032,-1046016,1597088,
%U -2418240,3632992,-5417708,8022840,-11802176,17252928,-25070568,36223424,-52053760,74414412
%N Expansion of 1/theta_3(q)^2 in powers of q.
%C Euler transform of period 4 sequence [ -4,6,-4,2,...].
%D A. Cayley, A memoir on the transformation of elliptic functions, Collected Mathematical Papers. Vols. 1-13, Cambridge Univ. Press, London, 1889-1897, Vol. 9, p. 128.
%H Vincenzo Librandi, <a href="/A004403/b004403.txt">Table of n, a(n) for n = 0..1000</a>
%H A. Cayley, <a href="/A001934/a001934.pdf">A memoir on the transformation of elliptic functions</a>, Philosophical Transactions of the Royal Society of London (1874): 397-456; Collected Mathematical Papers. Vols. 1-13, Cambridge Univ. Press, London, 1889-1897, included in Vol. 9. [Annotated scan of pages 126-129]
%F Expansion of (Sum x^(n^2), n = -inf .. inf )^(-2).
%F Expansion of elliptic function pi / 2K in powers of q.
%F G.f.: 1 / (Sum_{k} x^k^2)^2 = (Product_{k>0} (1 + x^(2k))^2 /((1-x^k)(1 + x^k)^3))^2.
%F a(n) = (-1)^n * A001934(n).
%t CoefficientList[Series[1/EllipticTheta[3, 0, q]^2, {q, 0, 32}], q] (* _Jean-François Alcover_, Jul 18 2011 *)
%t QP = QPochhammer; s = QP[q^2]^2/QP[-q]^4 + O[q]^40; CoefficientList[s, q] (* _Jean-François Alcover_, Nov 30 2015, adapted from PARI *)
%o (PARI) {a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A)^2 / eta(-x + A)^4, n))} /* _Michael Somos_, Feb 09 2006 */
%o (Julia) # JacobiTheta3 is defined in A000122.
%o A004403List(len) = JacobiTheta3(len, -2)
%o A004403List(33) |> println # _Peter Luschny_, Mar 12 2018
%Y Cf. A001934, A015128.
%K sign,easy
%O 0,2
%A _N. J. A. Sloane_
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