# Combinatorics Question involving compositions and polynomials

A composition of $k$ integers with sum $X$ is the set $S$ of all possible ordered sequence $(s_{1},s_{2},...,s_{k})$ of non-negative integers such that $s_{1}+s_{2}+...+s_{k}=X$. Many questions involving composition can be reduced to a generating-function problem. However, my question is, if we wished to know the nature of the polynomial (and perhaps explicitly find it)

$a_{0}+a_{1}z^{1}+a_{2}z^{2}...$, where $a_{i}$ is the number of sequences in $S$ such that $(-1)^{s_{1}}+(-1)^{s_{2}}+...+(-1)^{s_{k}}= i$.

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