A304627 a(n) = [x^n] Product_{k>=1} (1 + x^k)*(1 - x^(n*k))/((1 - x^k)*(1 + x^(n*k))).

1, 0, 2, 6, 12, 22, 38, 62, 98, 152, 230, 342, 502, 726, 1038, 1470, 2060, 2862, 3946, 5398, 7334, 9902, 13286, 17726, 23526, 31064, 40822, 53406, 69566, 90246, 116622, 150142, 192610, 246254, 313806, 398638, 504884, 637590, 802934, 1008446, 1263270, 1578526, 1967694, 2447062

Offset: 0

Author

Ilya Gutkovskiy, May 15 2018

Keywords

• nonn

Links

• Vaclav Kotesovec, Table of n, a(n) for n = 0..10000
• Eric Weisstein's World of Mathematics, Jacobi Theta Functions
• Index entries for sequences related to partitions

Crossrefs

Cf. A000065, A015128, A052839, A080054, A098151, A103258, A111133, A138526, A262968.

Programs

Mathematica

Table[SeriesCoefficient[Product[(1 + x^k) (1 - x^(n k))/((1 - x^k) (1 + x^(n k))) , {k, 1, n}], {x, 0, n}], {n, 0, 43}]
Table[SeriesCoefficient[Product[(1 + x^k)/(1 - x^k), {k, 1, n - 1}], {x, 0, n}], {n, 0, 43}]
Join[{1}, Table[SeriesCoefficient[EllipticTheta[4, 0, x^n]/EllipticTheta[4, 0, x], {x, 0, n}], {n, 43}]]
nmax = 43; CoefficientList[Series[1/EllipticTheta[4, 0, x] - 2 x/(1 - x), {x, 0, nmax}], x]

Formula

G.f.: 1/theta_4(x) - 2*x/(1 - x), where theta_4() is the Jacobi theta function.

a(n) ~ exp(Pi*sqrt(n)) / (8*n). - Vaclav Kotesovec, May 19 2018