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
KEYWORD
sign
AUTHOR
STATUS
approved