A004672 Theta series of extremal even unimodular lattice in dimension 48.

1, 0, 0, 52416000, 39007332000, 6609020221440, 437824977408000, 15173208925056000, 327259384199748000, 4913603518247424000, 55439899840480296960, 496425571825135680000, 3672747716246756784000

Offset

0,4

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, Equation 8, 7-8.

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.

Formula

G.f.: E4(q)^6 - 1440 * E4(q)^3 * Delta(q) + 125280 * Delta(q)^2 with E4(q) as in A004009 and Delta(q) as in A000594.

Example

G.f.: 1 + 52416000*q^3 + 39007332000*q^4 + ...

Prog

(Sage)
e4 = eisenstein_series_qexp(4, 20, normalization = "integral");
delta = CuspForms(1, 12).0.q_expansion(20);
e4^6-1440*e4^3 *delta+125280*delta^2 # Andy Huchala, May 09 2021

Crossrefs

Cf. A000594, A004009.

Sequence in context: A015392 A211237 A105381 * A029824 A186596 A205180

Adjacent sequences: A004669 A004670 A004671 * A004673 A004674 A004675

Keyword

nonn

Author

N. J. A. Sloane

Status

approved