On Wendt's determinant

Wendt's determinant of order $m$ is the circulant determinant $W_{m}$ whose $(i,j)$-th entry is the binomial coefficient $\binom m{|i-j|}$, for $1\leq i,j\leq m$. We give a formula for $W_{m}$, when $m$ is even not divisible by 6, in terms of the discriminant of a polynomial $T_{m+1}$, with rational coefficients, associated to $(X+1)^{m+1}-X^{m+1}-1$. In particular, when $m=p-1$ where $p$ is a prime $\equiv -1 (mod 6)$, this yields a factorization of $W_{p-1}$ involving a Fermat quotient, a power of $p$ and the 6-th power of an integer.