A376021 a(n) = Sum_{d|n} d^(n/d - d) * binomial(n/d-1,d-1).
1, 1, 1, 2, 1, 5, 1, 13, 2, 33, 1, 90, 1, 193, 55, 450, 1, 1295, 1, 2321, 1216, 5121, 1, 16528, 2, 24577, 20413, 54529, 1, 193446, 1, 254721, 295246, 524289, 376, 2254023, 1, 2359297, 3897235, 5329176, 1, 24303263, 1, 23986177, 48404882, 46137345, 1, 274687104, 2
Offset: 1
Keywords
Crossrefs
Cf. A376017.
Programs
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PARI
a(n) = sumdiv(n, d, d^(n/d-d)*binomial(n/d-1, d-1));
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PARI
my(N=50, x='x+O('x^N)); Vec(sum(k=1, N, (x^k/(1-k*x^k))^k)) -
Python
from math import comb from itertools import takewhile from sympy import divisors def A376021(n): return sum(d**((m:=n//d)-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} ( x^k / (1 - k*x^k) )^k.
If p is prime, a(p) = 1.