Signs behaviour of sums of weighted numbers of compositions

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Let A be a subset of positive integers. For a given positive integer n and 0≤i≤n, let cA(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 (SA,k(n))n∈N, where SA,k(n)=∑i=0n(-1)kikcA(i,n). We prove that for a broad class of subsets A, the number (-1)nSA,k(n) is non-negative for all sufficiently large n. Moreover, we show that there exists A⊂N+ such that the sign behaviour of SA,k(n) is not periodic.