If one factorizes by CAS the expression $$x^{\frac{m(m+1)}{2}}\prod_{k=1}^m(x^k+(\frac{1}{x})^k)$$ a puzzling perfect factorization seems to be possible for all natural values of $m$.
E.g. for $m=5$ I got this factorization with the cyclotomic polynomials $\Phi_n(u)$ $$\Phi_2^3(x^2)\Phi_4(x^2)\Phi_6(x^2)\Phi_8(x^2)\Phi_{10}(x^2)$$
For $m=8$ this factorization was calculated in a similar way $$\Phi_2^4(x^2)\Phi_4^2(x^2)\Phi_6(x^2)\Phi_8(x^2)\Phi_{10}(x^2)\Phi_{12}(x^2)\Phi_{14}(x^2)\Phi_{16}(x^2)$$
The exponent of $\Phi_2(x^2)$ seems to be $\frac m2$ for even $m$ and $\frac{m+1}{2}$ for odd $n$.
Interesting is the evaluation at $x=e^{\frac{\pi i}{m}}$.
It would be interesting how e.g. the exponents of the cyclotomic polynomials in the factorization appear and are distributed when $m\to\infty$.
Can a general law be calculated by pencil-and-paper method ? I see no way how to tackle this.
