Signs behaviour of sums of weighted numbers of compositions

Let $A$ be a subset of positive integers. For a given positive integer $n$ and $0\le i\le n$ let $c_A(i,n)$ denotes 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 is $A\subset {\mathbb{N}}_{+}$ such that the sign behaviour of $S_{A,k}(n)$ is not periodic.