  We study cluster algebras with principal and arbitrary coefficient systems
that are associated to unpunctured surfaces. We give a direct formula for the
Laurent polynomial expansion of cluster variables in these cluster algebras in
terms of certain paths on a triangulation of the surface. As an immediate
consequence, we prove the positivity conjecture of Fomin and Zelevinsky for
these cluster algebras.
  Furthermore, we obtain direct formulas for F-polynomials and g-vectors and
show that F-polynomials have constant term equal to 1. As an application, we
compute the Euler-Poincar\'e characteristic of quiver Grassmannians in Dynkin
type $A$ and affine Dynkin type $\tilde A$.

Representation Theory;Rings and Algebras