Factors of $x^n+1$ over $\mathbb{Z}[x]$

Question

Is there any equivalent to

$x^n-1 = \prod\limits_{d|n} \phi_d$ where $\phi_d$ is the $d$th cyclotomic polynomial

but for $x^n+1$?

Even better, can we generalize any further?

Answer 1

Yes. In fact :

$$ X^n+1=\frac{X^{2n}-1}{X^n-1}=\prod_{d|2n, d\not| n} \Phi_d $$