A002482 Theta series of Borcherds' 27-dimensional unimodular lattice U_27.

1, 0, 0, 2664, 101142, 1645056, 16045056, 110146176, 584713404, 2549741568, 9515943936, 31314087864, 92917622376, 252775586304, 638328674304, 1511740886400, 3387163161366, 7228598851584

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn

Links

• R. Bacher and B. B. Venkov, Réseaux entiers unimodulaires sans racine en dimension 27 et 28, in Réseaux euclidiens, designs sphériques et formes modulaires, pp. 212-267, Enseignement Math., Geneva, 2001.
• J. H. Conway and N. J. A. Sloane, Sphere Packings, Lattices and Groups, Springer-Verlag, Preface to 3rd ed. [alternative link]
• 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.

Programs

Maple

th3^27-54*th3^19*delta8+216*th3^11*delta8^2 (th3 = A000122, delta8 = A002408).

Mathematica

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 *)