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

0,2

Links

Vaclav Kotesovec, Table of n, a(n) for n = 0..10000

Eric Weisstein's World of Mathematics, Jacobi Theta Functions

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.

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]

Crossrefs

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

Sequence in context: A166869 A168380 A335730 * A297184 A218871 A121569

Adjacent sequences: A308283 A308284 A308285 * A308287 A308288 A308289

Keyword

nonn

Author

Ilya Gutkovskiy, May 18 2019

Status

approved