Let $\alpha:\mathbb Z\longrightarrow \mathbb Z$ be a quadratic polynomial taking only integral values on the integers and consider the sequence of square-matrices with coefficients $x^{\alpha(i+j)}$ for $i,j\in \{0,\dots,n\}$. The determinant of such a matrix involves a (not nessecarily positive) power of $x$ and a polynomial which factors experimentally always into a product of cyclotomic polynomials. Is this always true or is there a counterexample to this observation?

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You can always factor out $x^{\alpha(i)}$ from each row and $x^{\alpha(j)}$ from each column, then multiply by $x^{n\alpha(0)}$ and you reduce to $\det(x^{\beta ij})=\prod (x^{\beta i}-x^{\beta j})$ by Vandermonde, where $\beta$ is some integer.