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
Keywords
Links
- Vaclav Kotesovec, Table of n, a(n) for n = 0..10000
- Eric Weisstein's World of Mathematics, Jacobi Theta Functions
Crossrefs
Programs
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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.