A072839 Expansion of F_9(q^2).

1, 0, 0, 0, 0, 0, 0, 0, 18, 0, 0, 0, 0, 0, 72, 0, 0, 0, 240, 0, 252, 0, 0, 0, 0, 0, 504, 0, 0, 0, 0, 0, 1026, 0, 0, 0, 2160, 0, 1512, 0, 0, 0, 0, 0, 2664, 0, 0, 0, 0, 0, 3528, 0, 0, 0, 6720, 0, 5616, 0, 0, 0, 0, 0, 6552, 0, 0, 0, 0, 0, 9828, 0, 0, 0, 17520, 0, 11232, 0, 0, 0, 0, 0, 16380, 0, 0

Offset: 0

Author

N. J. A. Sloane, Jul 25 2002

Keywords

• nonn

Comments

Theta series of {A_8}* lattice. - Andy Huchala, Jul 01 2021

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000
• S. Ahlgren, The sixth, eighth, ninth and tenth powers of Ramanujan's theta function, Proc. Amer. Math. Soc. 128 (2000), 1333-1338.
• K. S. Chua. The Root Lattice An* and Ramanujan's Circular Summation of Theta Functions, Proceedings of the American Mathematical Society, 130 (2001), 1-8.

Crossrefs

Cf. A008448 (dual), A072835.

A023920 aerated with 0's.

Programs

Magma

L := Dual(Lattice("A", 8));
T := ThetaSeries(L,32); Coefficients(T); // Andy Huchala, Jul 01 2021

Mathematica

f[x_,y_]:= QPochhammer[-x, x*y]*QPochhammer[-y, x*y]*QPochhammer[x*y, x*y]; CoefficientList[Series[f[q^9, q^9]^8 - 16*q^9*f[q^9, q^27]^8 + 256*q^18*f[q^18, q^54]^8 + 18*q^8*f[q^18, -q^36]^12/f[q^6, -q^12]^4, {q, 0, 100}], q] (* G. C. Greubel, Apr 15 2018 *)