Consider the sequence

$$ a(n) = \prod_{u^n=1,u \neq 1}( (1+u)^n+1) $$

Some terms are: $$ 1,1,0,9,121,2704,118336, 4092529,0,97734390625, \ldots $$

Alonso del Arte asks:

Question: What are the multiples of $3$ such that

$$ a(3k) =0 $$

I tried some factorization of cyclotomic polynomials without success. May be true for all odd $k$ ???

EDIT: Another simple property of the sequence is

(hope this may please the negative voter (???))

$$ a(p) \equiv 1 \pmod{p} $$

for any prime $p>3$

since

$$ a(n) (2^n+1) $$ is the determinant of a circulant matrix with first line $$ 3,\binom{n}{1}, \ldots,\binom{n-1}{n} $$