A023919 Theta series of A*_7 lattice. Expansion of F_8(q^2).

1, 0, 0, 0, 0, 0, 0, 16, 0, 0, 0, 0, 56, 0, 0, 112, 126, 0, 0, 0, 0, 0, 0, 336, 0, 0, 0, 0, 576, 0, 0, 672, 756, 0, 0, 0, 0, 0, 0, 1232, 0, 0, 0, 0, 1512, 0, 0, 2016, 2072, 0, 0, 0, 0, 0, 0, 2800, 0, 0, 0, 0, 4032, 0, 0, 4048, 4158, 0, 0, 0, 0, 0, 0, 5712, 0, 0, 0, 0, 5544, 0, 0, 6944, 7560

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn

Examples

			G.f. = 1 + 16*q^7 + 56*q^12 + 112*q^15 + 126*q^16 + 336*q^23 + 576*q^28 + 672*q^31 + 756*q^32 + 1232*q^39 + 1512*q^44 + 2016*q^47 + 2072*q^48 + O(q^49)
		

References

• J. H. Conway and N. J. A. Sloane, "Sphere Packings, Lattices and Groups", Springer-Verlag, p. 114.

Links

• G. C. Greubel, Table of n, a(n) for n = 0..10000
• S. Ahlgren, The sixth, eighth, ninth and tenth powers of Ramanujan's theta function, Proc. Amer. Math. Soc., 128 (1999), 1333-1338.
• G. Nebe and N. J. A. Sloane, Home page for this lattice

Crossrefs

Cf. A008447 (A_7).

Programs

Magma

L:=Lattice("A",7); D:=Dual(L); T1 := ThetaSeries(D,60);

Mathematica

terms = 81; phi[q_] := EllipticTheta[3, 0, q]; psi[q_] := (1/2)*q^(-1/8) * EllipticTheta[2, 0, q^(1/2)]; F8[q_] := (1/8) (phi[q^2]^7 + (2 Sqrt[q] psi[q^4])^7 + 14 phi[q^2]^5 phi[q]^2 - 7 phi[q^2]^3 phi[q]^4); s = Simplify[Normal[F8[q^2] + O[q]^terms], q>0]; CoefficientList[s, q][[1 ;; terms]] (* Jean-François Alcover, Jul 04 2017 *)