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A002433 Theta series of unique 26-dimensional unimodular lattice T_26 with no roots (and minimal norm 3). 1

%I #16 Oct 16 2023 23:24:03

%S 1,0,0,3120,102180,1482624,13191360,83859360,416587860,1712638720,

%T 6061945344,19019791440,54048571200,141266958720,343675612800,

%U 786321725280,1706284712340,3532676509440,7012626150400,13413721342320,24829712546184,44601384921600

%N Theta series of unique 26-dimensional unimodular lattice T_26 with no roots (and minimal norm 3).

%D R. E. Borcherds, The Leech Lattice and Other Lattices, Ph. D. Dissertation, Cambridge Univ., 1984.

%D J. H. Conway and N. J. A. Sloane, "Sphere Packings, Lattices and Groups", Springer-Verlag, Third Ed., pp. xli-xlii.

%H Vaclav Kotesovec, <a href="/A002433/b002433.txt">Table of n, a(n) for n = 0..1000</a>

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

%F Let f = theta_3, g = 8-dimensional cusp form [Conway-Sloane, p. 187, Eqs. (32)-(34)]. The theta-series is f^26 - 52*f^18*g + 156*f^10*g^2.

%e 1 + 3120*q^3 + 102180*q^4 + 1482624*q^5 + 13191360*q^6 + 83859360*q^7 + 416587860*q^8 + ...

%t terms = 22; QP = QPochhammer; f = EllipticTheta[3, 0, q]; g = q*(QP[q]*(QP[q^4]/QP[q^2]))^8; s = f^26 - 52*f^18*g + 156*f^10*g^2 + O[q]^terms; CoefficientList[s, q] (* _Jean-François Alcover_, Jul 06 2017 *)

%K nonn

%O 0,4

%A _N. J. A. Sloane_

%E Edited by _N. J. A. Sloane_, Aug 23 2008 at the suggestion of _R. J. Mathar_

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