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A320078
Expansion of Product_{k>0} theta_3(q^(2*k-1)), where theta_3() is the Jacobi theta function.
10
1, 2, 0, 2, 6, 2, 4, 6, 8, 16, 8, 14, 26, 26, 24, 30, 58, 50, 60, 78, 90, 118, 104, 138, 192, 224, 204, 268, 366, 354, 412, 474, 596, 694, 724, 818, 1052, 1162, 1176, 1470, 1756, 1918, 2052, 2434, 2814, 3168, 3396, 3806, 4674, 5124, 5396, 6250, 7374, 7898, 8732
OFFSET
0,2
LINKS
Vaclav Kotesovec, Table of n, a(n) for n = 0..10000 (terms 0..1000 from Seiichi Manyama)
Eric Weisstein's World of Mathematics, Jacobi Theta Functions
FORMULA
Expansion of Product_{k>0} eta(q^(2*(2*k-1)))^5 / (eta(q^(2*k-1))*eta(q^(4*(2*k-1))))^2.
a(n) ~ (log(2))^(1/4) * exp(Pi*sqrt(n*log(2)/2)) / (4*n^(3/4)). - Vaclav Kotesovec, Oct 07 2018
MATHEMATICA
nmax = 60; CoefficientList[Series[Product[EllipticTheta[3, 0, x^(2*k-1)], {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Oct 05 2018 *)
nmax = 60; CoefficientList[Series[Product[(1 - x^((2*k-1)*j))*(1 + x^((2*k-1)*j))^3/(1 + x^(2*j*(2*k-1)))^2, {k, 1, nmax}, {j, 1, Floor[nmax/k] + 1}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Oct 06 2018 *)
PROG
(PARI) q='q+O('q^80); Vec(prod(k=1, 50, eta(q^(2*(2*k-1)))^5/( eta(q^(2*k-1))* eta(q^(4*(2*k-1))))^2 ) ) \\ G. C. Greubel, Oct 29 2018
CROSSREFS
KEYWORD
nonn
AUTHOR
Seiichi Manyama, Oct 05 2018
STATUS
approved