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A308286 Expansion of Product_{i>=1, j>=1} theta_3(x^(i*j)), where theta_3() is the Jacobi theta function.

%I A308286 #10 Feb 16 2025 08:33:55
%S A308286 1,2,4,12,20,40,84,140,252,456,752,1260,2128,3392,5436,8760,13582,
%T A308286 21092,32744,49620,75104,113448,168508,249620,368840,538412,783480,
%U A308286 1136652,1634000,2341280,3344680,4743684,6706120,9452392,13245800,18504888,25777520,35735376
%N A308286 Expansion of Product_{i>=1, j>=1} theta_3(x^(i*j)), where theta_3() is the Jacobi theta function.
%H A308286 Vaclav Kotesovec, <a href="/A308286/b308286.txt">Table of n, a(n) for n = 0..10000</a>
%H A308286 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/JacobiThetaFunctions.html">Jacobi Theta Functions</a>
%F A308286 G.f.: Product_{k>=1} theta_3(x^k)^tau(k), where tau = number of divisors (A000005).
%F A308286 G.f.: Product_{i>=1, j>=1} (Sum_{k=-oo..+oo} x^(i*j*k^2)).
%F A308286 G.f.: Product_{i>=1, j>=1, k>=1} (1 - x^(i*j*k))*(1 + x^(i*j*k))^3/(1 + x^(2*i*j*k))^2.
%F A308286 G.f.: Product_{k>=1} (1 - x^k)^tau_3(k)*(1 + x^k)^(3*tau_3(k))/(1 + x^(2*k))^(2*tau_3(k)), where tau_3 = A007425.
%t A308286 nmax = 37; CoefficientList[Series[Product[Product[EllipticTheta[3, 0, x^(i j)], {j, 1, nmax}], {i, 1, nmax}], {x, 0, nmax}], x]
%t A308286 nmax = 37; CoefficientList[Series[Product[EllipticTheta[3, 0, x^k]^DivisorSigma[0, k], {k, 1, nmax}], {x, 0, nmax}], x]
%Y A308286 Cf. A000005, A000122, A006171, A007425, A300446, A301554, A305050, A308288, A320067, A320908, A321241.
%K A308286 nonn
%O A308286 0,2
%A A308286 _Ilya Gutkovskiy_, May 18 2019