A319672 a(n) = [x^n] Product_{k>=2} ((1 + x^k)/(1 - x^k))^n.

1, 0, 4, 6, 40, 110, 520, 1778, 7568, 28320, 116224, 453046, 1837600, 7306234, 29565848, 118786526, 481192480, 1945153838, 7895908852, 32046260282, 130370798320, 530650047710, 2163191769336, 8824509524082, 36037768384832, 147277910160160, 602398740105712, 2465582764631334

Offset

0,3

Links

Vaclav Kotesovec, Table of n, a(n) for n = 0..1000

Formula

a(n) = [x^n] ((1 - x)/((1 + x)*theta_4(x)))^n, where theta_4() is the Jacobi theta function.

a(n) = [x^n] exp(n*Sum_{k>=1} (sigma(2*k) - sigma(k) + (-1)^k - 1)*x^k/k).

a(n) ~ c * d^n / sqrt(n), where d = 4.16958962845360844086951404338054148667024... and c = 0.23380422010834870751549442953816486722... - Vaclav Kotesovec, Oct 06 2018

Mathematica

Table[SeriesCoefficient[Product[((1 + x^k)/(1 - x^k))^n , {k, 2, n}], {x, 0, n}], {n, 0, 27}]
Table[SeriesCoefficient[((1 - x)/((1 + x) EllipticTheta[4, 0, x]))^n, {x, 0, n}], {n, 0, 27}]
Table[SeriesCoefficient[Exp[n Sum[(DivisorSigma[1, 2 k] - DivisorSigma[1, k] + (-1)^k - 1) x^k/k, {k, 1, n}]], {x, 0, n}], {n, 0, 27}]

Crossrefs

Cf. A000203, A270919, A300415, A319670, A319671.

Sequence in context: A175349 A281223 A023644 * A355233 A377682 A145387

Adjacent sequences: A319669 A319670 A319671 * A319673 A319674 A319675

Keyword

nonn

Author

Ilya Gutkovskiy, Sep 25 2018

Status

approved