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%I #32 Jul 06 2021 20:18:25
%S 1,0,146880,64757760,4844836800,137695887360,2121555283200,
%T 21421110804480,158757684004800,928986331545600,4512164186816640,
%U 18847854517248000,69519016873985280,230952108679004160
%N Theta series of extremal even unimodular lattice in dimension 32.
%C There are at least 15 such lattices, one of which is the Barnes-Wall lattice BW_32.
%D J. H. Conway and N. J. A. Sloane, "Sphere Packings, Lattices and Groups", Springer-Verlag, p. 195.
%H Andy Huchala, <a href="/A004670/b004670.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>
%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.
%H G. Nebe and N. J. A. Sloane, <a href="http://www.math.rwth-aachen.de/~Gabriele.Nebe/LATTICES/KV32RM.html">Home page for this lattice</a>
%H <a href="/index/Ba#BW">Index entries for sequences related to Barnes-Wall lattices</a>
%F a(n) = A282012(n) - 960*A027364(n-1) for n > 0. - _Andy Huchala_, May 01 2021
%e G.f.: 1 + 146880*q^2 + 64757760*q^3 + 4844836800*q^4 + ...
%o (Sage)
%o e4 = eisenstein_series_qexp(4,20,normalization = "integral");
%o delta = CuspForms(1,12).0.q_expansion(20);
%o (e4^4 - 960*delta*e4).list()[:20] # _Andy Huchala_, May 01 2021
%Y Cf. A027364, A282012.
%K nonn
%O 0,3
%A _N. J. A. Sloane_
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