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 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). 3

%I

%S 1,0,144,-5120,70524,-626688,4265600,-24164352,119375370,-529539072,

%T 2151757440,-8125793280,28827864296,-96885780480,310514729472,

%U -954123868160,2823202073655,-8074060259328,22387521828480,-60344692402176,158484892943628,-406368240128000,1019049374174976

%N 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).

%H Seiichi Manyama, <a href="/A319294/b319294.txt">Table of n, a(n) for n = -2..10000</a>

%H M. Viazovska, <a href="https://arxiv.org/abs/1603.04246">The sphere packing problem in dimension 8</a>, arXiv preprint arXiv:1603.04246 [math.NT], 2016.

%e Let q = exp(Pi i t).

%e theta_2(q)^4 = 16*q + 64*q^3 + ... .

%e theta_3(q)^4 = 1 + 8*q + 24*q^2 + 32*q^3 + ... .

%e theta_4(q)^4 = 1 - 8*q + 24*q^2 - 32*q^3 + ... .

%e 128 * (theta_3(q)^4 + theta_4(q)^4)/theta_2(q)^8

%e = q^(-2) + 16 - 132*q^2 + ... .

%e 128 * (theta_4(q)^4 - theta_2(q)^4)/theta_3(q)^8

%e = 128 - 5120*q + 70656*q^2 - ... .

%e G.f.: q^(-2) + 144 - 5120*q + 70524*q^2 - 626688*q^3 + 4265600*q^4 - 24164352*q^5 + ... .

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

%K sign

%O -2,3

%A _Seiichi Manyama_, Sep 16 2018

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Last modified April 11 11:59 EDT 2021. Contains 342886 sequences. (Running on oeis4.)