Irreducible factorization of translates of reversed Dickson polynomials over finite fields

Let $F$ be a field of $q$ elements, where $q$ is a power of an odd prime. Fix $n=(q+1)\mathrm{/}2$. For each $s\in F$, we describe all the irreducible factors over $F$ of the polynomial $g_s(y):=y^n+(1-y{)}^n-s$, and we give a necessary and sufficient condition on $s$ for $g_s(y)$ to be irreducible.