A319294 Expansion of 128 * ((theta_3(q)^4 + theta_4(q)^4)/theta_2(q)^8 + (theta_4(q)^4 - theta_2(q)^4)/theta_3(q)^8) in powers of q = exp(Pi i t).

1, 0, 144, -5120, 70524, -626688, 4265600, -24164352, 119375370, -529539072, 2151757440, -8125793280, 28827864296, -96885780480, 310514729472, -954123868160, 2823202073655, -8074060259328, 22387521828480, -60344692402176, 158484892943628, -406368240128000, 1019049374174976

Offset

-2,3

Links

Seiichi Manyama, Table of n, a(n) for n = -2..10000

Maryna S. Viazovska, The sphere packing problem in dimension 8, arXiv preprint arXiv:1603.04246 [math.NT], 2016.

Example

Let q = exp(Pi i t).
theta_2(q)^4 =     16*q          + 64*q^3 + ... .
theta_3(q)^4 = 1 +  8*q + 24*q^2 + 32*q^3 + ... .
theta_4(q)^4 = 1 -  8*q + 24*q^2 - 32*q^3 + ... .
128 * (theta_3(q)^4 + theta_4(q)^4)/theta_2(q)^8
= q^(-2) + 16 - 132*q^2 + ... .
128 * (theta_4(q)^4 - theta_2(q)^4)/theta_3(q)^8
= 128 - 5120*q + 70656*q^2 - ... .
G.f.: q^(-2) + 144 - 5120*q + 70524*q^2 - 626688*q^3 + 4265600*q^4 - 24164352*q^5 + ... .

Crossrefs

Cf. A000118 (theta_3(q)^4), A008438 (theta_2(q)^4/(16*q)), A096727 (theta_4(q)^4), A281373.

Sequence in context: A008376 A221127 A262788 * A233645 A086985 A371251

Adjacent sequences: A319291 A319292 A319293 * A319295 A319296 A319297

Keyword

sign

Author

Seiichi Manyama, Sep 16 2018

Status

approved