  By definition, a first return is the immediate moment that a path, using
vectors in the Cartesian plane, touches the $x$-axis after leaving it
previously from a given point; the initial point is often the origin. In this
case, using certain diagonal and horizontal vectors while restricting the
movements to the first quadrant will cause almost every first return to end at
the point $(2n,0)$, where $2n$ counts the equal number of up and down steps in
a path. The exception will be explained further in the sections below.
  Using the first returns of Catalan, Schr\"{o}der, and Motzkin numbers, which
resulted from the lattice paths formed using a combination of diagonal and/or
horizontal vectors, we then investigated the effect that coloring select
vectors will have on each of the original generating functions.
