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Expansion of Product_{k>0} theta_3(q^(2*k))/theta_3(q^(2*k-1)), where theta_3() is the Jacobi theta function.
2

%I #16 Oct 26 2018 17:09:59

%S 1,-2,6,-14,28,-58,114,-210,384,-684,1178,-2010,3372,-5538,9006,

%T -14466,22906,-35954,55884,-85946,131176,-198622,298274,-444958,

%U 659368,-970544,1420362,-2066876,2990680,-4305598,6168154,-8793554,12480718,-17637250,24818530,-34785622

%N Expansion of Product_{k>0} theta_3(q^(2*k))/theta_3(q^(2*k-1)), where theta_3() is the Jacobi theta function.

%H Vaclav Kotesovec, <a href="/A321027/b321027.txt">Table of n, a(n) for n = 0..10000</a> (terms 0..1000 from Seiichi Manyama)

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/JacobiThetaFunctions.html">Jacobi Theta Functions</a>

%t nmax = 50; CoefficientList[Series[Product[EllipticTheta[3, 0, x^(2*k)] / EllipticTheta[3, 0, x^(2*k-1)], {k, 1, nmax}], {x, 0, nmax}], x] (* _Vaclav Kotesovec_, Oct 26 2018 *)

%Y Cf. A000122, A320067, A320098, A321026.

%K sign

%O 0,2

%A _Seiichi Manyama_, Oct 26 2018