A main motivation for this paper is the search for the sufficient condition of the primality of an integer $n$ in order that the congruence $1^{(n-1)}+2^{(n-1)}+3^{(n-1)}+\cdots+(n-1)^{(n-1)} \equiv -1$ (mod $n$) holds. Some properties of Jacobi polynomials were investigated using certain Kummer results. Certain properties of Bernoulli polynomials as well as the Staudt-Clausen theorem for prime factors were also used. In this paper, several new properties of the coefficients of the polynomial $$ d(m,k)=(-1)^k\cdot\frac{2^k}{k}\cdotum_{s\geq\frac{k}{2}}^{k-1} \binom{2m-1}{2k-2s-1}\cdot \binom{2s-1}{k-1}\cdot B_{2m-2k+2s} $$, have been obtained, and they are formulated in Theorem 1 and Theorem 2.