A cluster expansion formula (A_n case)

We consider the Ptolemy cluster algebras, which are cluster algebras of finite type $A$ (with non-trivial coefficients) that have been described by Fomin and Zelevinsky using triangulations of a regular polygon. Given any seed $\zS$ in a Ptolemy cluster algebra, we present a formula for the expansion of an arbitrary cluster variable in terms of the cluster variables of the seed $\zS$. Our formula is given in a combinatorial way, using paths on a triangulation of the polygon that corresponds to the seed $\zS$.