A002433 Theta series of unique 26-dimensional unimodular lattice T_26 with no roots (and minimal norm 3).

1, 0, 0, 3120, 102180, 1482624, 13191360, 83859360, 416587860, 1712638720, 6061945344, 19019791440, 54048571200, 141266958720, 343675612800, 786321725280, 1706284712340, 3532676509440, 7012626150400, 13413721342320, 24829712546184, 44601384921600

Offset

0,4

References

R. E. Borcherds, The Leech Lattice and Other Lattices, Ph. D. Dissertation, Cambridge Univ., 1984.

J. H. Conway and N. J. A. Sloane, "Sphere Packings, Lattices and Groups", Springer-Verlag, Third Ed., pp. xli-xlii.

Links

Vaclav Kotesovec, Table of n, a(n) for n = 0..1000

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

Formula

Let f = theta_3, g = 8-dimensional cusp form [Conway-Sloane, p. 187, Eqs. (32)-(34)]. The theta-series is f^26 - 52*f^18*g + 156*f^10*g^2.

Example

1 + 3120*q^3 + 102180*q^4 + 1482624*q^5 + 13191360*q^6 + 83859360*q^7 + 416587860*q^8 + ...

Mathematica

terms = 22; QP = QPochhammer; f = EllipticTheta[3, 0, q]; g = q*(QP[q]*(QP[q^4]/QP[q^2]))^8; s = f^26 - 52*f^18*g + 156*f^10*g^2 + O[q]^terms; CoefficientList[s, q] (* Jean-François Alcover, Jul 06 2017 *)

Crossrefs

Sequence in context: A357957 A269324 A183850 * A107535 A181285 A133526

Adjacent sequences: A002430 A002431 A002432 * A002434 A002435 A002436

Keyword

nonn

Author

N. J. A. Sloane

Extensions

Edited by N. J. A. Sloane, Aug 23 2008 at the suggestion of R. J. Mathar

Status

approved