Stirling polynomials

An investigation is made of the polynomials f_k(n) = S(n + k, n) and g_k(n) = (?1)^ks(n, n ? k), where S and s denote the Stirling numbers of the second and first kind, respectively. The main result gives a combinatorial interpretation of the coefficients of the polynomial (1 ? x)^2k+1Σ_n=0^∞f_k(n)xⁿ analogous to the well-known combinatorial interpretation of the Eulerian numbers in terms of descents of permutations.