  Given a quiver associated to a cluster algebra and a sequence of vertices,
iterative mutation leads to $F$-Polynomials which appear in numerous places in
the cluster algebraic literature. The coefficients of the monomials in these
$F$-Polynomials are difficult to understand and have been an area of study for
many years. In this paper, we present a general closed-form formula for these
coefficients in terms of elementary manipulations with $C$-matrices. We then
demonstrate the effectiveness of the formula by using it to derive simple
explicit formulas for $F$-Polynomials of specific classes of quivers and
mutation sequences. Work has been done to do these cases in ad-hoc
combinatorial ways, but our formula recovers and improves known formulas with a
general method.
  Secondly, we investigate convergence of $F$-polynomials. In themselves, they
do not converge, but by changing bases using the $C$-matrix, they conjecturally
do. Since our formula relates $C$-matrix entries to coefficients, we are able
to apply it to make considerable progress on the conjecture. Specifically, we
show stability for green mutation sequences. Finally, we look at exact formulas
for these stable deformed $F$-polynomials in instances where they illustrate
properties that the $F$-polynomials themselves do not.
