cp's OEIS Frontend

A308288 Expansion of Product_{i>=1, j>=1} theta_3(x^(i*j))/theta_4(x^(i*j)), where theta_() is the Jacobi theta function.

%I A308288 #8 Feb 16 2025 08:33:55
%S A308288 1,4,16,56,172,496,1360,3528,8824,21344,50048,114360,255336,557888,
%T A308288 1195952,2519264,5221076,10660512,21467904,42674520,83812560,
%U A308288 162753584,312689168,594740456,1120498048,2092059800,3872731232,7110830376,12955269304,23428775520
%N A308288 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 A308288 Convolution of the sequences A305050 and A308286.
%H A308288 Vaclav Kotesovec, <a href="/A308288/b308288.txt">Table of n, a(n) for n = 0..10000</a>
%H A308288 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/JacobiThetaFunctions.html">Jacobi Theta Functions</a>
%F A308288 G.f.: Product_{k>=1} (theta_3(x^k)/theta_4(x^k))^tau(k), where tau = number of divisors (A000005).
%F A308288 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 A308288 G.f.: Product_{i>=1, j>=1, k>=1} (1 + x^(i*j*k))^4/(1 + x^(2*i*j*k))^2.
%F A308288 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 A308288 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 A308288 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 A308288 Cf. A000005, A000122, A002448, A007096, A007425, A301554, A305050, A308286, A320967, A320970.
%K A308288 nonn
%O A308288 0,2
%A A308288 _Ilya Gutkovskiy_, May 18 2019