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A004403
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Expansion of 1/theta_3(q)^2 in powers of q.
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2
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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
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,2
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COMMENTS
| Euler transform of period 4 sequence [ -4,6,-4,2,...].
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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.
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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).
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MATHEMATICA
| CoefficientList[Series[1/EllipticTheta[3, 0, q]^2, {q, 0, 32}], q] (* From Jean-François Alcover, Jul 18 2011 *)
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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 */
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CROSSREFS
| Cf. A001934, A015128.
Sequence in context: A127811 A138517 A001934 * A084566 A079769 A107035
Adjacent sequences: A004400 A004401 A004402 * A004404 A004405 A004406
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KEYWORD
| sign,easy
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AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com).
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