A369873 - OEIS

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

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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