A108094 - OEIS

%I #21 Oct 20 2018 10:23:01

%S 1,0,270,3840,-514080,-15413760,1283087040,62644907520,-3378279124350,

%T -252933976704000,8502815843769600,1007506223570707200,

%U -17757117956815481280,-3942183666885514421760,14527133705347401150720,15088544258811557869278720,144818514010649047069497600

%N Coefficients of series whose 16th power is the theta series of the 16-dimensional Barnes-Wall lattice (see A008409).

%H Seiichi Manyama, <a href="/A108094/b108094.txt">Table of n, a(n) for n = 0..551</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>, arXiv:math/0509316 [math.NT], 2005-2006.

%H N. Heninger, E. M. Rains and N. J. A. Sloane, <a href="http://dx.doi.org/10.1016/j.jcta.2006.03.018">On the Integrality of n-th Roots of Generating Functions</a>, J. Combinatorial Theory, Series A, 113 (2006), 1732-1745.

%e More precisely, the theta series of the Barnes-Wall lattice begins 1 + 4320*q^2 + 61440*q^3 + 522720*q^4 + 2211840*q^5 + 8960640*q^6 + 23224320*q^7 + ... and the 16th root of this is 1 + 270*q^2 + 3840*q^3 - 514080*q^4 - 15413760*q^5 + 1283087040*q^6 + 62644907520*q^7 - ...

%t f[q_] := 1/2 (EllipticTheta[2, 0, q]^16 + EllipticTheta[3, 0, q]^16 + EllipticTheta[4, 0, q]^16 + 30 EllipticTheta[2, 0, q]^8 EllipticTheta[3, 0, q]^8);

%t CoefficientList[f[q]^(1/16) + O[q]^17, q] (* _Jean-François Alcover_, Aug 17 2018 *)

%K sign

%O 0,3

%A _N. J. A. Sloane_ and _Michael Somos_, Jun 06 2005