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

0,2

Comments

Euler transform of period 4 sequence [ -4,6,-4,2,...].

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]

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).

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 *)

Prog

(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 */
(Julia) # JacobiTheta3 is defined in A000122.
A004403List(len) = JacobiTheta3(len, -2)
A004403List(33) |> println # Peter Luschny, Mar 12 2018

Crossrefs

Cf. A001934, A015128.

Sequence in context: A361099 A138517 A001934 * A084566 A208903 A079769

Adjacent sequences: A004400 A004401 A004402 * A004404 A004405 A004406

Keyword

sign,easy

Author

N. J. A. Sloane

Status

approved