%I #7 May 29 2019 14:13:08

%S 1,4,16,56,172,496,1360,3528,8824,21344,50048,114360,255336,557888,

%T 1195952,2519264,5221076,10660512,21467904,42674520,83812560,

%U 162753584,312689168,594740456,1120498048,2092059800,3872731232,7110830376,12955269304,23428775520

%N Expansion of Product_{i>=1, j>=1} theta_3(x^(i*j))/theta_4(x^(i*j)), where theta_() is the Jacobi theta function.

%C Convolution of the sequences A305050 and A308286.

%H Vaclav Kotesovec, <a href="/A308288/b308288.txt">Table of n, a(n) for n = 0..10000</a>

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

%F G.f.: Product_{k>=1} (theta_3(x^k)/theta_4(x^k))^tau(k), where tau = number of divisors (A000005).

%F G.f.: Product_{i>=1, j>=1} (Sum_{k=-oo..+oo} x^(i*j*k^2))/(Sum_{k=-oo..+oo} (-1)^k*x^(i*j*k^2)).

%F G.f.: Product_{i>=1, j>=1, k>=1} (1 + x^(i*j*k))^4/(1 + x^(2*i*j*k))^2.

%F G.f.: Product_{k>=1} (1 + x^k)^(4*tau_3(k))/(1 + x^(2*k))^(2*tau_3(k)), where tau_3 = A007425.

%t nmax = 29; CoefficientList[Series[Product[Product[EllipticTheta[3, 0, x^(i j)]/EllipticTheta[4, 0, x^(i j)], {j, 1, nmax}], {i, 1, nmax}], {x, 0, nmax}], x]

%t nmax = 29; CoefficientList[Series[Product[(EllipticTheta[3, 0, x^k]/EllipticTheta[4, 0, x^k])^DivisorSigma[0, k], {k, 1, nmax}], {x, 0, nmax}], x]

%Y Cf. A000005, A000122, A002448, A007096, A007425, A301554, A305050, A308286, A320967, A320970.

%K nonn

%O 0,2

%A _Ilya Gutkovskiy_, May 18 2019