An elementary proof of the symmetry of power sum polynomials

It is well known that the power sums 1ᵏ⁻¹+2ᵏ⁻¹+⋯+nᵏ⁻¹ are polynomials pₖ(n) of degree k in n . We provide an elementary proof of the fact that the polynomial pₖ(x) is symmetric with respect to the line x=-1/2 or the point (-1/2,0) , depending on whether k is even or odd, respectively.