A369874 a(n) is the constant term in the expansion of Product_{d|n} (x^d + 1 + 1/x^d).

1, 1, 1, 1, 1, 5, 1, 1, 1, 1, 1, 23, 1, 1, 1, 1, 1, 13, 1, 13, 1, 1, 1, 103, 1, 1, 1, 7, 1, 77, 1, 1, 1, 1, 1, 175, 1, 1, 1, 63, 1, 49, 1, 1, 5, 1, 1, 463, 1, 1, 1, 1, 1, 41, 1, 39, 1, 1, 1, 2975, 1, 1, 3, 1, 1, 33, 1, 1, 1, 25, 1, 2363, 1, 1, 1, 1, 1, 25, 1, 261

Offset: 1

Author

Ilya Gutkovskiy, Feb 03 2024

Keywords

• nonn

Comments

a(n) is the number of solutions to 0 = Sum_{d|n} c_i * d with c_i in {-1,0,1}, i=1..tau(n), tau = A000005.

Links

• Alois P. Heinz, Table of n, a(n) for n = 1..20000

Crossrefs

Cf. A000005, A007576, A033630, A083206, A369873.

Programs

Mathematica

Table[Coefficient[Product[(x^d + 1 + 1/x^d), {d, Divisors[n]}], x, 0], {n, 1, 80}]

Python

from collections import Counter
from sympy import divisors
def A369874(n):
    c = {0:1}
    for d in divisors(n,generator=True):
        b = Counter(c)
        for j in c:
            a = c[j]
            b[j+d] += a
            b[j-d] += a
        c = b
    return c[0] # Chai Wah Wu, Feb 05 2024