A004403 Expansion of 1/theta_3(q)^2 in powers of q.
1, -4, 12, -32, 76, -168, 352, -704, 1356, -2532, 4600, -8160, 14176, -24168, 40512, -66880, 108876, -174984, 277932, -436640, 679032, -1046016, 1597088, -2418240, 3632992, -5417708, 8022840, -11802176, 17252928, -25070568, 36223424, -52053760, 74414412
Offset: 0
References
- 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.
Links
- Vincenzo Librandi, Table of n, a(n) for n = 0..1000
- A. Cayley, A memoir on the transformation of elliptic functions, 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]
Programs
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Julia
# JacobiTheta3 is defined in A000122. A004403List(len) = JacobiTheta3(len, -2) A004403List(33) |> println # Peter Luschny, Mar 12 2018
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Mathematica
CoefficientList[Series[1/EllipticTheta[3, 0, q]^2, {q, 0, 32}], q] (* Jean-François Alcover, Jul 18 2011 *) 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 *) -
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 */
Formula
Expansion of (Sum x^(n^2), n = -inf .. inf )^(-2).
Expansion of elliptic function pi / 2K in powers of q.
G.f.: 1 / (Sum_{k} x^k^2)^2 = (Product_{k>0} (1 + x^(2k))^2 /((1-x^k)(1 + x^k)^3))^2.
a(n) = (-1)^n * A001934(n).
Comments