A327249 Expansion of Sum_{k>=1} x^k * (1 + k * x^k)^k.

1, 2, 1, 5, 1, 14, 1, 17, 28, 26, 1, 160, 1, 50, 251, 321, 1, 622, 1, 1607, 1030, 122, 1, 6257, 3126, 170, 2917, 12202, 1, 27291, 1, 28929, 6656, 290, 84036, 117721, 1, 362, 13183, 407121, 1, 417881, 1, 220100, 850312, 530, 1, 2246465, 823544, 2100626

Offset: 1

Author

Ilya Gutkovskiy, Sep 15 2019

Keywords

• nonn

Crossrefs

Cf. A006005 (positions of 1's), A087909, A217668, A260180, A327238.

Programs

Magma

[&+[(n div d)^(d-1)*Binomial(n div d,d-1):d in Divisors(n)]:n in [1..50]]; // Marius A. Burtea, Sep 15 2019

Mathematica

nmax = 50; CoefficientList[Series[Sum[x^k (1 + k x^k)^k, {k, 1, nmax}], {x, 0, nmax}], x] // Rest
Table[DivisorSum[n, (n/#)^(# - 1) Binomial[n/#, # - 1] &], {n, 1, 50}]

PARI

a(n) = sumdiv(n, d, (n/d)^(d-1) * binomial(n/d,d-1)); \\ Michel Marcus, Sep 15 2019

Formula

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