A004672 - OEIS

%I #18 Oct 26 2023 00:59:08

%S 1,0,0,52416000,39007332000,6609020221440,437824977408000,

%T 15173208925056000,327259384199748000,4913603518247424000,

%U 55439899840480296960,496425571825135680000,3672747716246756784000

%N Theta series of extremal even unimodular lattice in dimension 48.

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

%H Andy Huchala, <a href="/A004672/b004672.txt">Table of n, a(n) for n = 0..20000</a>

%H N. Elkies, <a href="http://people.math.harvard.edu/~elkies/M272.19/nov04.pdf">Rational Lattices and their Theta Functions</a>, Equation 8, 7-8.

%H N. Heninger, E. M. Rains and N. J. A. Sloane, <a href="https://arxiv.org/abs/math/0509316">On the Integrality of n-th Roots of Generating Functions</a>, arXiv:math/0509316 [math.NT], 2005-2006; J. Combinatorial Theory, Series A, 113 (2006), 1732-1745.

%F 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.

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

%o (Sage)

%o e4 = eisenstein_series_qexp(4,20,normalization = "integral");

%o delta = CuspForms(1,12).0.q_expansion(20);

%o e4^6-1440*e4^3 *delta+125280*delta^2 # _Andy Huchala_, May 09 2021

%Y Cf. A000594, A004009.

%K nonn

%O 0,4

%A _N. J. A. Sloane_