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

1, 2, 4, 12, 20, 40, 84, 140, 252, 456, 752, 1260, 2128, 3392, 5436, 8760, 13582, 21092, 32744, 49620, 75104, 113448, 168508, 249620, 368840, 538412, 783480, 1136652, 1634000, 2341280, 3344680, 4743684, 6706120, 9452392, 13245800, 18504888, 25777520, 35735376

Offset: 0

Author

Ilya Gutkovskiy, May 18 2019

Keywords

• nonn

Links

• Vaclav Kotesovec, Table of n, a(n) for n = 0..10000
• Eric Weisstein's World of Mathematics, Jacobi Theta Functions

Crossrefs

Cf. A000005, A000122, A006171, A007425, A300446, A301554, A305050, A308288, A320067, A320908, A321241.

Programs

Mathematica

nmax = 37; CoefficientList[Series[Product[Product[EllipticTheta[3, 0, x^(i j)], {j, 1, nmax}], {i, 1, nmax}], {x, 0, nmax}], x]
nmax = 37; CoefficientList[Series[Product[EllipticTheta[3, 0, x^k]^DivisorSigma[0, k], {k, 1, nmax}], {x, 0, nmax}], x]

Formula

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

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

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.

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.