A002483 Expansion of Jacobi theta function {theta_1}'(q) in powers of q^(1/4).

0, 2, 0, 0, 0, 0, 0, 0, 0, -6, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 10, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -14, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 18, 0, 0, 0

Offset

0,2

References

J. Tannery and J. Molk, Eléments de la Théorie des Fonctions Elliptiques, Vol. 2, Gauthier-Villars, Paris, 1902; Chelsea, NY, 1972, see p. 27.

E. T. Whittaker and G. N. Watson, A Course of Modern Analysis, Cambridge Univ. Press, 4th ed., 1963, p. 464.

Links

Antti Karttunen, Table of n, a(n) for n = 0..10201

J. H. Conway and N. J. A. Sloane, Sphere Packings, Lattices and Groups, Springer-Verlag, p. 102.

Formula

Expansion of 2 * q^(-1/8) * eta(q)^3 in powers of q. - Michael Somos, May 31 2012

G.f.: 2 * x * Product_{k>0} (1 - x^(8*k))^3. - Michael Somos, May 31 2012

For n > 0, a(n) = (((1/8)*(4*t^2 + 4*t + 1 - n) - 1)*4 + 2)*(t-r)*(-1)^(t+1), where t = floor((sqrt(n)+1)/2) and r = floor((sqrt(n-1)+1)/2). - Mikael Aaltonen, Jan 16 2015

Example

2*x - 6*x^9 + 10*x^25 - 14*x^49 + 18*x^81 - 22*x^121 + 26*x^169 - 30*x^225 + ...

Maple

Sum( (-1)^m*(2*m+1)*q^ ( ((2*m+1)/2)^2 ), m=-10, 10);

Mathematica

a[ n_] := SeriesCoefficient[ EllipticThetaPrime[ 1, 0, q], {q, 0, n/4}] (* Michael Somos, May 31 2012 *)
s = 2q*QPochhammer[q^8]^3+O[q]^90; CoefficientList[s, q] (* Jean-François Alcover, Nov 30 2015, adapted from PARI *)

Prog

(PARI) {a(n) = local(m); if( issquare( n, &m) && m%2, 2 * (-1)^(m \ 2) * m, 0)} /* Michael Somos, May 31 2012 */
(PARI) {a(n) = if( n<1, 0, n--; polcoeff( 2 * eta(x^8 + x * O(x^n))^3, n))} /* Michael Somos, May 31 2012 */

Crossrefs

Dividing by 2 gives (essentially) A245552.

See also the g.f. for A053187.

Sequence in context: A359431 A338210 A122698 * A282530 A060478 A088806

Adjacent sequences: A002480 A002481 A002482 * A002484 A002485 A002486

Keyword

sign

Author

N. J. A. Sloane

Status

approved