A366104 G.f. ( Chi(sqrt(x))^4 + Chi(-sqrt(x))^4 )/2, where Chi(x) = Product_{k >= 0} 1 + x^(2*k+1) is the g.f. of A000700.
1, 6, 17, 38, 84, 172, 325, 594, 1049, 1796, 3005, 4912, 7877, 12430, 19309, 29580, 44766, 66978, 99150, 145374, 211242, 304382, 435194, 617674, 870651, 1219352, 1697283, 2348888, 3232919, 4426546, 6030872, 8177986, 11039633, 14838518, 19862613, 26482878, 35175989, 46552818, 61393694
Offset: 0
Programs
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Maple
with(QDifferenceEquations): seq(coeff((1/2)*expand(QPochhammer(-q,q^2,40)^4 + QPochhammer(q,q^2,40)^4), q, 2*n), n = 0..40); #alternative program seq(coeff(expand(QPochhammer(-q^2, q^2, 20)^2 * QPochhammer(-q, q^2, 20)^6), q, n), n = 0..40);
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Mathematica
nmax = 40; CoefficientList[Series[Product[(1 + x^(2*k))^2 * (1 + x^(2*k-1))^6, {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Jun 29 2025 *)
Formula
G.f.: Product_{k >= 1} (1 + x^(2*k))^2*(1 + x^(2*k-1))^6.
G.f.: x^(1/12) * eta(x^2)^10 * eta(x^4)^2 / ( eta(x) * eta(x^4) )^6.
a(n) ~ exp(2*Pi*sqrt(n/3)) / (4 * 3^(1/4) * n^(3/4)). - Vaclav Kotesovec, Jun 29 2025
Comments