  In this note, we find an explicit formula for the Laurent expression of
cluster variables of coefficient-free rank two cluster algebras associated with
the matrix $\left(\begin{array}{cc} 0 & c -c & 0 \end{array}\right)$, and show
that a large number of coefficients are non-negative. As a corollary, we obtain
an explicit expression for the Euler-Poincar\'{e} characteristics of the
corresponding quiver Grassmannians.
