Polynomial Formula for Sums of Powers of Integers

In this article, it is shown that for any positive integer k >= 1, there exist unique real numbers a(kr), r = 1, 2, ..., (k + 1), such that for any integer n >= 1 S-k,S-n equivalent to Sigma(n)(j=1) j(k) = Sigma((k+1))(r=1) a(kr)n(r). The numbers a(kr) are computed explicitly for r = k + 1, k, k - 1, ..., (k - 10). This fully determines the polynomials for k = 1, 2, ..., 12. The cases k = 1, 2, 3 are well known and available in high school algebra books.