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A317643
Expansion of theta_3(q^3)*theta_3(q^4), where theta_3() is the Jacobi theta function.
0
1, 0, 0, 2, 2, 0, 0, 4, 0, 0, 0, 0, 2, 0, 0, 0, 6, 0, 0, 4, 0, 0, 0, 0, 0, 0, 0, 2, 4, 0, 0, 4, 0, 0, 0, 0, 2, 0, 0, 4, 0, 0, 0, 4, 0, 0, 0, 0, 6, 0, 0, 0, 4, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 4, 6, 0, 0, 4, 0, 0, 0, 0, 0, 0, 0, 2, 4, 0, 0, 4, 0, 0, 0, 0, 4, 0, 0, 0, 0, 0, 0, 8, 0, 0, 0, 0, 0, 0, 0, 0, 2
OFFSET
0,4
COMMENTS
Number of integer solutions to the equation 3*x^2 + 4*y^2 = n.
LINKS
N. J. A. Sloane et al., Binary Quadratic Forms and OEIS (Index to related sequences, programs, references)
Eric Weisstein's World of Mathematics, Jacobi Theta Functions
FORMULA
G.f.: Product_{k>=1} (1 + x^(6*k-3))^2*(1 - x^(6*k))*(1 + x^(8*k-4))^2*(1 - x^(8*k)).
EXAMPLE
G.f. = 1 + 2*q^3 + 2*q^4 + 4*q^7 + 2*q^12 + 6*q^16 + 4*q^19 + 2*q^27 + 4*q^28 + ...
MATHEMATICA
nmax = 100; CoefficientList[Series[EllipticTheta[3, 0, q^3] EllipticTheta[3, 0, q^4], {q, 0, nmax}], q]
nmax = 100; CoefficientList[Series[QPochhammer[-q^3, -q^3] QPochhammer[-q^4, -q^4]/(QPochhammer[q^3, -q^3] QPochhammer[q^4, -q^4]), {q, 0, nmax}], q]
CROSSREFS
KEYWORD
nonn
AUTHOR
Ilya Gutkovskiy, Aug 02 2018
STATUS
approved