A320078 - OEIS

%I #28 Feb 16 2025 08:33:56

%S 1,2,0,2,6,2,4,6,8,16,8,14,26,26,24,30,58,50,60,78,90,118,104,138,192,

%T 224,204,268,366,354,412,474,596,694,724,818,1052,1162,1176,1470,1756,

%U 1918,2052,2434,2814,3168,3396,3806,4674,5124,5396,6250,7374,7898,8732

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

%H Vaclav Kotesovec, <a href="/A320078/b320078.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="https://mathworld.wolfram.com/JacobiThetaFunctions.html">Jacobi Theta Functions</a>

%F Expansion of Product_{k>0} eta(q^(2*(2*k-1)))^5 / (eta(q^(2*k-1))*eta(q^(4*(2*k-1))))^2.

%F a(n) ~ (log(2))^(1/4) * exp(Pi*sqrt(n*log(2)/2)) / (4*n^(3/4)). - _Vaclav Kotesovec_, Oct 07 2018

%t nmax = 60; CoefficientList[Series[Product[EllipticTheta[3, 0, x^(2*k-1)], {k, 1, nmax}], {x, 0, nmax}], x] (* _Vaclav Kotesovec_, Oct 05 2018 *)

%t nmax = 60; CoefficientList[Series[Product[(1 - x^((2*k-1)*j))*(1 + x^((2*k-1)*j))^3/(1 + x^(2*j*(2*k-1)))^2, {k, 1, nmax}, {j, 1, Floor[nmax/k] + 1}], {x, 0, nmax}], x] (* _Vaclav Kotesovec_, Oct 06 2018 *)

%o (PARI) q='q+O('q^80); Vec(prod(k=1,50, eta(q^(2*(2*k-1)))^5/( eta(q^(2*k-1))* eta(q^(4*(2*k-1))))^2 ) ) \\ _G. C. Greubel_, Oct 29 2018

%Y Cf. A000122, A033716, A320067, A320098.

%K nonn,changed

%O 0,2

%A _Seiichi Manyama_, Oct 05 2018