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A320067
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Expansion of Product_{k>0} theta_3(q^k), where theta_3() is the Jacobi theta function.
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21
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1, 2, 2, 6, 8, 10, 22, 26, 36, 60, 78, 106, 152, 202, 258, 370, 478, 602, 828, 1042, 1332, 1758, 2198, 2758, 3572, 4448, 5518, 7012, 8636, 10654, 13350, 16362, 19946, 24722, 30070, 36478, 44776, 54010, 65202, 79234, 95196, 114166, 137686, 164530, 196252, 235308, 279718, 332002
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OFFSET
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0,2
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COMMENTS
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Also the number of integer solutions (a_1, a_2, ..., a_n) to the equation a_1^2 + 2*a_2^2 + ... + n*a_n^2 = n.
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LINKS
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FORMULA
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Expansion of Product_{k>0} eta(q^(2*k))^5 / (eta(q^k)*eta(q^(4*k)))^2.
a(n) ~ log(2)^(3/8) * exp(Pi*sqrt(n*log(2))) / (4 * Pi^(1/4) * n^(7/8)). - Vaclav Kotesovec, Oct 05 2018
Expansion of Product_{k>0} theta_4(q^(2*k))/theta_4(q^(2*k-1)), where theta_4() is the Jacobi theta function. - Seiichi Manyama, Oct 26 2018
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MATHEMATICA
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nmax = 50; CoefficientList[Series[Product[EllipticTheta[3, 0, x^k], {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Oct 05 2018 *)
nmax = 50; CoefficientList[Series[Product[(1 - x^(k*j))*(1 + x^(k*j))^3/(1 + x^(2*k*j))^2, {k, 1, nmax}, {j, 1, Floor[nmax/k] + 1}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Oct 05 2018 *)
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PROG
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(PARI) m=50; x='x+O('x^m); Vec(1/(prod(k=1, 2*m, prod(j=1, floor(2*m/k), (1 - x^(k*j))*(1 + x^(k*j))^3/(1 + x^(2*k*j))^2 )))) \\ G. C. Greubel, Oct 29 2018
(Magma) m:=50; R<x>:=PowerSeriesRing(Integers(), m); Coefficients(R!((&*[(&*[(1 - x^(k*j))*(1 + x^(k*j))^3/(1 + x^(2*k*j))^2: j in [1..Floor(2*m/k)]]): k in [1..2*m]]))); // G. C. Greubel, Oct 29 2018
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CROSSREFS
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KEYWORD
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nonn,nice
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AUTHOR
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STATUS
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approved
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