cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-2 of 2 results.

A319670 a(n) = [x^n] Product_{k>=2} 1/(1 - x^k)^n.

Original entry on oeis.org

1, 0, 2, 3, 14, 30, 119, 301, 1078, 3036, 10242, 30624, 100451, 310128, 1004817, 3158343, 10182982, 32345186, 104145896, 332953929, 1072383374, 3442913407, 11100120528, 35742258497, 115377720235, 372326184555, 1203406838428, 3890040945078, 12588182588373, 40748118469180
Offset: 0

Views

Author

Ilya Gutkovskiy, Sep 25 2018

Keywords

Comments

Number of partitions of n into parts > 1, if there are n kinds of parts.

Crossrefs

Programs

  • Mathematica
    Table[SeriesCoefficient[Product[1/(1 - x^k)^n , {k, 2, n}], {x, 0, n}], {n, 0, 29}]
    Table[SeriesCoefficient[((1 - x)/QPochhammer[x])^n, {x, 0, n}], {n, 0, 29}]
    Table[SeriesCoefficient[Exp[n Sum[(DivisorSigma[1, k] - 1) x^k/k, {k, 1, n}]], {x, 0, n}], {n, 0, 29}]

Formula

a(n) = [x^n] exp(n*Sum_{k>=1} (sigma(k) - 1)*x^k/k).
a(n) ~ c * d^n / sqrt(n), where d = 3.293558598422332665054219310876308... and c = 0.2154241499279313950113565475... - Vaclav Kotesovec, Oct 06 2018

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

Original entry on oeis.org

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

Views

Author

Ilya Gutkovskiy, Sep 25 2018

Keywords

Crossrefs

Programs

  • 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}]

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
Showing 1-2 of 2 results.