A324158 Expansion of Sum_{k>=1} x^k / (1 - k * x^k)^k.

1, 2, 2, 6, 2, 23, 2, 50, 56, 107, 2, 660, 2, 499, 1592, 2370, 2, 8246, 2, 18557, 21786, 11387, 2, 175198, 43752, 53419, 298892, 487762, 2, 1891098, 2, 2552066, 3905222, 1114403, 3785462, 29081597, 2, 4981099, 48376512, 95510772, 2, 218764940, 2, 346411232, 770590352

Offset: 1

Author

Ilya Gutkovskiy, Sep 02 2019

Keywords

• nonn

Links

• Seiichi Manyama, Table of n, a(n) for n = 1..5000

Crossrefs

Cf. A055225, A087909, A157019, A157020, A167531, A324159.

Programs

Mathematica

nmax = 45; CoefficientList[Series[Sum[x^k/(1 - k x^k)^k, {k, 1, nmax}], {x, 0, nmax}], x] // Rest
Table[Sum[(n/d)^(d - 1) Binomial[n/d + d - 2, d - 1], {d, Divisors[n]}], {n, 1, 45}]

PARI

a(n) = sumdiv(n, d, (n/d)^(d-1) * binomial(n/d+d-2, d-1)); \\ Michel Marcus, Sep 02 2019

PARI

N=66; x='x+O('x^N); Vec(sum(k=1, N, x^k/(1-k*x^k)^k)) \\ Seiichi Manyama, Sep 03 2019

Formula

a(n) = Sum_{d|n} (n/d)^(d-1) * binomial(n/d+d-2,d-1).

a(p) = 2, where p is prime.