  We prove a closed character formula for the symmetric powers $S^N V(\lambda)$
of a fixed irreducible representation $V(\lambda)$ of a complex semi-simple Lie
algebra $\mathfrak{g}$ by means of partial fraction decomposition. The formula
involves rational functions in rank of $\mathfrak{g}$ many variables which are
easier to determine than the weight multiplicities of $S^N V(\lambda)$
themselves. We compute those rational functions in some interesting cases.
Furthermore, we introduce a residue-type generating function for the weight
multiplicities of $S^N V(\lambda)$ and explain the connections between our
character formula, vector partition functions and iterated partial fraction
decomposition.
