A008424 Theta series of {D_9}* lattice.

1, 0, 0, 0, 18, 0, 0, 0, 144, 512, 0, 0, 672, 0, 0, 0, 2034, 4608, 0, 0, 4320, 0, 0, 0, 7392, 18432, 0, 0, 12672, 0, 0, 0, 22608, 47616, 0, 0, 34802, 0, 0, 0, 44640, 101376, 0, 0, 60768, 0, 0, 0, 93984, 193536, 0, 0, 125280, 0, 0, 0, 141120, 324096, 0, 0

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• easy

Examples

			G.f. = 1 + 18*q^4 + 144*q^8 + ...
		

References

• J. H. Conway and N. J. A. Sloane, "Sphere Packings, Lattices and Groups", Springer-Verlag, p. 120.

Links

• Andy Huchala, Table of n, a(n) for n = 0..3000
• G. Nebe and N. J. A. Sloane, Home page for this lattice

Crossrefs

Cf. A008431.

Programs

Magma

L := Dual(Lattice("D", 9));
B := Basis(ThetaSeriesModularFormSpace(L), 100);
S := [ 1, 0, 0, 0, 18];
Coefficients(&+[B[i] * S[i] : i in [1..5]]); // Andy Huchala, Jul 24 2021

PARI

N=66;  q='q+O('q^N);
T3(q) = eta(q^2)^5 / ( eta(q)^2 * eta(q^4)^2 );
T2(q) = eta(q^4)^2 / eta(q^2);
Vec( T3(q^4)^9 + (2 * q * T2(q^4))^9 )
\\ Joerg Arndt, Mar 29 2018

Formula

Theta series in terms of Jacobi theta series: (theta_2)^9 + (theta_3)^9. - Sean A. Irvine, Mar 28 2018

Extensions

More terms from Andy Huchala, Jul 24 2021