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 (|i-j|^m), for 1 ≤ i, j ≤ 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+i, with rational coefficients, associated to (X + 1)^m+1 - X^m+1 - 1. In particular, when m = p - 1 where p is a prime ≡ - 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.