> 
> Given a positive integer $n$, express$$
> f\_n(x) = \left|\begin{array}{c c c c c} 
> 1 & x & \cdots & x^{n - 1} & x^n\\
> x & 1 & x & \cdots & x^{n - 1} \\
> \vdots & x & \ddots & \ddots & \vdots\\
> x^{n - 1} & \vdots & \ddots & 1 & x\\
> x^n & x^{n - 1} & \cdots & x & 1
> \end{array}\right|
> $$
>  as a polynomial of $x$.
> 
> 
> 

I tried to find a recurrence relation of $\{f\_n\}\_{n \geqslant 1}$ using Laplace expansion, but there seems to be no patterns in the minors in the expansion. Is there a somewhat simple recurrence relation of $\{f\_n\}\_{n \geqslant 1}$ or these determinants can be computed with other methods?