A004670 Theta series of extremal even unimodular lattice in dimension 32.

1, 0, 146880, 64757760, 4844836800, 137695887360, 2121555283200, 21421110804480, 158757684004800, 928986331545600, 4512164186816640, 18847854517248000, 69519016873985280, 230952108679004160

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn

Comments

There are at least 15 such lattices, one of which is the Barnes-Wall lattice BW_32.

Examples

			G.f.: 1 + 146880*q^2 + 64757760*q^3 + 4844836800*q^4 + ...
		

References

• J. H. Conway and N. J. A. Sloane, "Sphere Packings, Lattices and Groups", Springer-Verlag, p. 195.

Links

• Andy Huchala, Table of n, a(n) for n = 0..20000
• N. Elkies, Rational Lattices and their Theta Functions
• N. Heninger, E. M. Rains and N. J. A. Sloane, On the Integrality of n-th Roots of Generating Functions, arXiv:math/0509316 [math.NT], 2005-2006; 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

Crossrefs

Cf. A027364, A282012.

Programs

Sage

e4 = eisenstein_series_qexp(4,20,normalization = "integral");
delta = CuspForms(1,12).0.q_expansion(20);
(e4^4 - 960*delta*e4).list()[:20] # Andy Huchala, May 01 2021

Formula

a(n) = A282012(n) - 960*A027364(n-1) for n > 0. - Andy Huchala, May 01 2021