A239917 - OEIS

%I #26 May 15 2023 08:45:43

%S 1,0,0,512,4320,18432,61440,193536,522720,1126400,2211840,4584960,

%T 8960640,14764032,23224320,40221696,67154400,96546816,135168000,

%U 210332160,319809600,423976960,550195200,801119232,1147643520,1436147712,1771683840,2462397440,3371915520

%N Theta series of 16-dimensional lattice OBW16, an overlattice of the Barnes-Wall lattice BW16.

%C The 512 vectors of norm 3 form a spherical 5-design (see Neumaier, 1981). The corresponding configuration of 256 lines in 16-space was studied by Shult and Yanushka, 1980.

%C This theta series is an element of the space of modular forms on Gamma_0(4) of weight 8 and dimension 5. - _Andy Huchala_, May 15 2023

%H Andy Huchala, <a href="/A239917/b239917.txt">Table of n, a(n) for n = 0..10000</a>

%H G. Nebe and N. J. A. Sloane, <a href="http://www.math.rwth-aachen.de/~Gabriele.Nebe/LATTICES/obw16.html">Home page for this lattice</a>

%H A. Neumaier, <a href="https://www.mat.univie.ac.at/~neum/scan/combcon.pdf">Combinatorial configurations in terms of distances</a>, Report 81-09-Wiskunde, Tech. Univ. Eindhoven, 1981.

%H A. Neumaier, <a href="http://dx.doi.org/10.1137/0604017">Lattices of simplex type</a>, SIAM J. Algebraic Discrete Methods 4 (1983), no. 2, 145--160. MR0699768 (85f:05040). See Example 3.

%H Ernest Shult, Arthur Yanushka, Arthur, <a href="http://dx.doi.org/10.1007/BF00156473">Near n-gons and line systems</a>, Geom. Dedicata 9 (1980), no. 1, 1--72. MR0566437 (82b:51018).

%e The theta series is 1 + 512*q^3 + 4320*q^4 + 18432*q^5 + 61440*q^6 + 193536*q^7 + 522720*q^8 + 1126400*q^9 + 2211840*q^10 + 4584960*q^11 + 8960640*q^12 + 14764032*q^13 + 23224320*q^14 + 40221696*q^15 + 67154400*q^16 + O(q^17).

%o (Magma)

%o L:=LatticeWithGram(16, [3,

%o -1,3,

%o -1,-1,3,

%o 1,-1,1,3,

%o 0,1,0,-1,3,

%o -1,0,0,-1,-1,3,

%o 1,0,0,1,-1,-1,3,

%o -1,0,1,0,1,-1,-1,3,

%o 1,0,-1,0,-1,1,1,-1,3,

%o -1,0,1,0,-1,0,0,1,-1,3,

%o 0,1,0,1,0,0,1,-1,0,0,3,

%o 1,-1,1,1,-1,0,1,-1,0,0,0,3,

%o -1,0,1,1,0,-1,0,1,-1,1,0,0,3,

%o 0,0,1,0,1,0,-1,1,-1,0,0,0,-1,3,

%o 1,1,-1,0,0,0,0,-1,0,0,1,0,-1,1,3,

%o 1,-1,-1,1,-1,-1,1,0,0,0,0,0,0,-1,0,3]);

%o T<q>:=ThetaSeries(L,16);

%o T;

%K nonn

%O 0,4

%A _N. J. A. Sloane_, Apr 13 2014

%E More terms from _Andy Huchala_, May 15 2023