In this note, we revisit the problem of polynomial interpolation and explicitly construct two polynomials in n of degree k + 1, P[subscript k](n) and Q[subscript k](n), such that P[subscript k](n) = Q[subscript k](n) = f[subscript k](n) for n = 1, 2,… , k, where f[subscript k](1), f[subscript k](2),… , f[subscript k](k) are k arbitrarily chosen (real or complex) values. Then, we focus on the case that fk(n) is given by the sum of powers of the first n positive integers S[subscript k](n) = 1[superscript k] + 2[superscript k] + ··· + n[superscript k], and show that S[subscript k](n) admits the polynomial representations S[subscript k](n) = P[subscript k](n) and S[subscript k](n) = Q[subscript k](n) for all n = 1, 2,… , and k = 1, where the first representation involves the Eulerian numbers, and the second one the Stirling numbers of the second kind. Finally, we consider yet another polynomial formula for S[subscript k](n) alternative to the well-known formula of Bernoulli.