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%I #31 Oct 16 2023 23:31:05
%S 1,0,0,2664,101142,1645056,16045056,110146176,584713404,2549741568,
%T 9515943936,31314087864,92917622376,252775586304,638328674304,
%U 1511740886400,3387163161366,7228598851584
%N Theta series of Borcherds' 27-dimensional unimodular lattice U_27.
%H R. Bacher and B. B. Venkov, <a href="https://www-fourier.univ-grenoble-alpes.fr/?q=fr/content/reseaux-entiers-unimodulaires-sans-racine-en-dimension-27-et-28">Réseaux entiers unimodulaires sans racine en dimension 27 et 28</a>, in Réseaux euclidiens, designs sphériques et formes modulaires, pp. 212-267, Enseignement Math., Geneva, 2001.
%H J. H. Conway and N. J. A. Sloane, <a href="https://doi.org/10.1007/978-1-4757-2016-7">Sphere Packings, Lattices and Groups</a>, Springer-Verlag, Preface to 3rd ed. [<a href="https://www.researchgate.net/publication/46957856_Sphere_Packings_Lattices_and_Groups">alternative link</a>]
%H N. Heninger, E. M. Rains and N. J. A. Sloane, <a href="http://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.
%p th3^27-54*th3^19*delta8+216*th3^11*delta8^2 (th3 = A000122, delta8 = A002408).
%t terms = 18; QP = QPochhammer; th3 = EllipticTheta[3, 0, q]; delta8 = q*(QP[q]*(QP[q^4]/QP[q^2]))^8; s = th3^27 - 54*th3^19*delta8 + 216*th3^11*delta8^2 + O[q]^terms; CoefficientList[s, q] (* _Jean-François Alcover_, Jul 06 2017 *)
%K nonn
%O 0,4
%A _N. J. A. Sloane_
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