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.

1, 4, 16, 56, 172, 496, 1360, 3528, 8824, 21344, 50048, 114360, 255336, 557888, 1195952, 2519264, 5221076, 10660512, 21467904, 42674520, 83812560, 162753584, 312689168, 594740456, 1120498048, 2092059800, 3872731232, 7110830376, 12955269304, 23428775520

Offset: 0

Author

Ilya Gutkovskiy, May 18 2019

Keywords

• nonn

Comments

Convolution of the sequences A305050 and A308286.

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, A002448, A007096, A007425, A301554, A305050, A308286, A320967, A320970.

Programs

Mathematica

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]
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]

Formula

G.f.: Product_{k>=1} (theta_3(x^k)/theta_4(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))/(Sum_{k=-oo..+oo} (-1)^k*x^(i*j*k^2)).

G.f.: Product_{i>=1, j>=1, k>=1} (1 + x^(i*j*k))^4/(1 + x^(2*i*j*k))^2.

G.f.: Product_{k>=1} (1 + x^k)^(4*tau_3(k))/(1 + x^(2*k))^(2*tau_3(k)), where tau_3 = A007425.