It is well known that the power sums 1^k-1+2^k-1++n^k-1 are polynomials p₊ (n) of degree k in n. We provide an elementary proof of the fact that the polynomial p₊ (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.
Helmut Länger (Tue,) studied this question.