A008409 Theta series of 16-dimensional Barnes-Wall lattice.
1, 0, 4320, 61440, 522720, 2211840, 8960640, 23224320, 67154400, 135168000, 319809600, 550195200, 1147643520, 1771683840, 3371915520, 4826603520, 8593797600, 11585617920, 19590534240, 25239859200, 40979580480, 50877235200
Offset: 0
Examples
1 + 4320*q^4 + 61440*q^6 + 522720*q^8 + 2211840*q^10 + 8960640*q^12 + ...
References
- J. H. Conway and N. J. A. Sloane, "Sphere Packings, Lattices and Groups", Springer-Verlag, p. 130, p. 131 Equation (132).
Links
- Seiichi Manyama, Table of n, a(n) for n = 0..10000 (terms 0..1000 from Vincenzo Librandi)
- N. Heninger, E. M. Rains and N. J. A. Sloane, On the Integrality of n-th Roots of Generating Functions, J. Combinatorial Theory, Series A, 113 (2006), 1732-1745.
- G. Nebe and N. J. A. Sloane, Home page for this lattice
- Index entries for sequences related to Barnes-Wall lattices
- Eric Weisstein's World of Mathematics, Theta Series
- Eric Weisstein's World of Mathematics, Barnes-Wall Lattice
Programs
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Mathematica
f[q_] := 1/2*(EllipticTheta[2, 0, q]^16 + EllipticTheta[3, 0, q]^16 + EllipticTheta[4, 0, q]^16 + 30*EllipticTheta[2, 0, q]^8*EllipticTheta[3, 0, q]^8); Series[f[q], {q, 0, 21}] // CoefficientList[#, q]& (* Jean-François Alcover, May 15 2013 *) -
PARI
{a(n) = local(A1, A2) ; if( n<0, 0, A1 = eta(x + x * O(x^n))^8; A2 = eta(x^2 + x * O(x^n))^8; polcoeff( (A1^6 + 32 * x * A1^3 * A2^3 + 4096 * x^2 * A2^6) / ( A1 * A2 )^2, n))} /* Michael Somos, Nov 29 2007 */
Formula
Expansion of ( theta_2(q)^16 + theta_3(q)^16 + theta_4(q)^16 + 30 * theta_2(q)^8 * theta_3(q)^8 ) / 2 in powers of q. - [Conway and Sloane]
Expansion of E_4(q^2)^2 + (E_4(q) - E_4(q^2))^2 / 15 in powers of q. - Michael Somos, Nov 29 2007
Expansion of ( eta(q)^48 + 32 * eta(q)^24 * eta(q^2)^24 + 4096 * eta(q^2)^48 ) / ( eta(q) * eta(q^2) )^16 in powers of q. - Michael Somos, Nov 29 2007
G.f. is Fourier series of a weight 8 level 2 modular form. f(-1 / (2 t)) = 16 (t/i)^8 f(t) where q = exp(2 Pi i t). - Michael Somos, Nov 29 2007