A320068 - OEIS

%I #36 Sep 08 2022 08:46:23

%S 1,-2,2,-6,12,-18,30,-50,84,-132,198,-306,476,-706,1026,-1522,2234,

%T -3202,4564,-6506,9224,-12934,17982,-24982,34612,-47496,64798,-88340,

%U 119944,-161814,217462,-291562,389642,-518442,687222,-908934,1199040,-1575730,2064466,-2699378,3520540

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

%H Vaclav Kotesovec, <a href="/A320068/b320068.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>

%F Convolution inverse of A320067.

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

%F Expansion of Product_{k>0} theta_4(q^(2*k-1))/theta_4(q^(2*k)), where theta_4() is the Jacobi theta function. - _Seiichi Manyama_, Oct 26 2018

%t nmax = 50; CoefficientList[Series[1/Product[EllipticTheta[3, 0, x^k], {k, 1, nmax}], {x, 0, nmax}], x] (* _Vaclav Kotesovec_, Oct 05 2018 *)

%t nmax = 50; CoefficientList[Series[Product[(1 + x^(2*k*j))^2/((1 - x^(k*j))*(1 + x^(k*j))^3), {k, 1, nmax}, {j, 1, Floor[nmax/k] + 1}], {x, 0, nmax}], x] (* _Vaclav Kotesovec_, Oct 08 2018 *)

%o (PARI) m=50; x='x+O('x^m); Vec(prod(k=1,2*m, prod(j=1,floor(2*m/k)+1, (1 + x^(2*k*j))^2/((1 - x^(k*j))*(1 + x^(k*j))^3) ))) \\ _G. C. Greubel_, Oct 29 2018

%o (Magma) m:=50; R<x>:=PowerSeriesRing(Integers(), m); Coefficients(R!((&*[(&*[(1 + x^(2*k*j))^2/((1 - x^(k*j))*(1 + x^(k*j))^3): j in [1..(Floor(2*m/k)+1)]]): k in [1..2*m]]))); // _G. C. Greubel_, Oct 29 2018

%Y Cf. A000122, A004402, A320067, A320069, A320070, A320968, A320992.

%K sign

%O 0,2

%A _Seiichi Manyama_, Oct 05 2018