A369873 a(n) is the constant term in the expansion of Product_{d|n} (x^d + 1/x^d).

0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 6, 0, 0, 0, 2, 0, 6, 0, 0, 0, 0, 0, 0, 0, 0, 0, 4, 0, 4, 0, 0, 0, 0, 0, 10, 0, 0, 0, 0, 0, 4, 0, 2, 0, 0, 0, 34, 0, 0, 0, 0, 0, 4, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 4, 0, 6, 0, 0, 0, 28, 0, 0, 0, 2, 0, 26, 0, 0, 0, 0, 0, 22, 0, 0, 0, 0, 0, 4, 0, 2, 0

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 {-1,1}, i=1..tau(n), tau = A000005.

Links

• Antti Karttunen, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A000005, A033630, A063865, A083206, A083207, A379504.

Programs

Mathematica

Table[Coefficient[Product[(x^d + 1/x^d), {d, Divisors[n]}], x, 0], {n, 1, 90}]

PARI

A369873(n) = { my(s=sigma(n),p=1); if(s%2 || s < 2*n, 0, fordiv(n, d, p *= ('x^d + 'x^-d)); polcoeff(p, 0)); }; \\ (cf. also code in A083206 and A379504) - Antti Karttunen, Jan 20 2025

Python

from collections import Counter
from sympy import divisors
def A369873(n):
    c = {0:1}
    for d in divisors(n,generator=True):
        b = Counter()
        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

Formula

From Joerg Arndt, Feb 04 2024: (Start)

a(n) != 0 (only) for n in A083207.

a(n) = 2 * A083206(n). (End)

Extensions

Data section extended to a(105) by Antti Karttunen, Jan 20 2025