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

0,2

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

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.

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]

Crossrefs

Cf. A000005, A000122, A002448, A007096, A007425, A301554, A305050, A308286, A320967, A320970.

Sequence in context: A115108 A127393 A239988 * A340257 A261386 A073388

Adjacent sequences: A308285 A308286 A308287 * A308289 A308290 A308291

Keyword

nonn

Author

Ilya Gutkovskiy, May 18 2019

Status

approved