A376020 a(n) = Sum_{d|n} d^(n/d) * binomial(n/d-1,d-1).

1, 1, 1, 5, 1, 17, 1, 49, 28, 129, 1, 564, 1, 769, 1459, 2049, 1, 11387, 1, 13313, 32806, 20481, 1, 223798, 3126, 98305, 551125, 540673, 1, 2662642, 1, 3276801, 7971616, 2097153, 1171876, 48412424, 1, 9437185, 105225319, 121675204, 1, 416694239, 1, 591396865

Offset: 1

Author

Seiichi Manyama, Sep 06 2024

Keywords

• nonn

Crossrefs

Cf. A376016.

Programs

PARI

a(n) = sumdiv(n, d, d^(n/d)*binomial(n/d-1, d-1));

PARI

my(N=50, x='x+O('x^N)); Vec(sum(k=1, N, (k*x^k/(1-k*x^k))^k))

Python

from math import comb
from itertools import takewhile
from sympy import divisors
def A376020(n): return sum(d**(m:=n//d)*comb(m-1,d-1) for d in takewhile(lambda d:d**2<=n,divisors(n))) # Chai Wah Wu, Sep 06 2024

Formula

G.f.: Sum_{k>=1} ( k*x^k / (1 - k*x^k) )^k.

If p is prime, a(p) = 1.