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
KEYWORD
sign,easy
AUTHOR
STATUS
approved