"I was somewhat surprised to see that the coefficients of the theta polynomial of the first shell of the E8 lattice -- namely 1,56,126,56,1 -- do not feature in the OEIS."
Adam, I agree - please go ahead and submit it - and others like it that should be in the OEIS. Not every shell in every lattice, of course! But a handful (or two) would be welcomed. Best regards Neil Neil J. A. Sloane, President, OEIS Foundation. 11 South Adelaide Avenue, Highland Park, NJ 08904, USA. Also Visiting Scientist, Math. Dept., Rutgers University, Piscataway, NJ. Phone: 732 828 6098; home page: http://NeilSloane.com Email: njasloane@gmail.com On Fri, Jan 19, 2018 at 7:34 AM, Adam P. Goucher <apgoucher@gmx.com> wrote:
It has occurred to me that (vertex-transitive) spherical codes have `theta polynomials' in the same way that lattices have `theta series'. Specifically, the coefficient of q^n gives the number of points at a Euclidean distance of sqrt(n).
I was somewhat surprised to see that the coefficients of the theta polynomial of the first shell of the E8 lattice -- namely 1,56,126,56,1 -- do not feature in the OEIS.
When you only have three non-zero coefficients, you can define a regular graph on the vertices of the spherical code. If the spherical code has sufficiently many symmetries -- specifically, that the point stabiliser of one vertex is transitive on each of the 'shells' whose cardinalities are specified in the coefficients of the polynomial -- then this graph is strongly regular. The standard construction of the Higman-Sims graph comes from such a spherical code [living in the Leech lattice].
Best wishes,
Adam P. Goucher