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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. 2
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 (list; graph; refs; listen; history; text; internal format)
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 * A261386 A073388 A109634

Adjacent sequences:  A308285 A308286 A308287 * A308289 A308290 A308291

KEYWORD

nonn

AUTHOR

Ilya Gutkovskiy, May 18 2019

STATUS

approved

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Last modified January 17 14:57 EST 2020. Contains 330958 sequences. (Running on oeis4.)