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

Offset

1,6

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

Table of n, a(n) for n=1..90.

Formula

From Joerg Arndt, Feb 04 2024: (Start)

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

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

Mathematica

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

Prog

(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

Crossrefs

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

Sequence in context: A089798 A070536 A318886 * A030201 A349797 A055668

Adjacent sequences: A369870 A369871 A369872 * A369874 A369875 A369876

Keyword

nonn

Author

Ilya Gutkovskiy, Feb 03 2024

Status

approved