Let A be a subset of positive integers. For a given positive integer n and \(0 \le i \le n\) , let \(c_{A}(i, n)\) denote the number of A-compositions of n with exactly i parts. In this note, we investigate the sign behaviour of the sequence \((S_{A, k}(n))_{n \in \mathbb {N}}\) , where \(S_{A, k}(n) = \sum _{i=0}^{n} (-1)^{k} i^{k} c_{A}(i, n)\) . We prove that for a broad class of subsets A, the number \((-1)^{n} S_{A, k}(n)\) is non-negative for all sufficiently large n. Moreover, we show that there exists \(A \subset \mathbb {N}_{+}\) such that the sign behaviour of \(S_{A, k}(n)\) is not periodic.
