  Polytope numbers for a given polytope are an integer sequence defined by the
combinatorics of the polytope. Recent work by H. K. Kim and J. Y. Lee has
focused on writing polytope number sequences as sums of simplex number
sequences. In addition, these works have given a process for writing the
polytope number sequence in a recursive fashion by using the interior sequence
for the various k-faces of the polytope, each viewed as a k-dimensional
polytope. This paper shows that the coefficients of the linear combination of
simplex number are the h-vector components for a certain type of triangulation
of the polytope. In addition, reversing the order of the coefficients in the
linear combination is shown to equal the interior polytope sequence for this
polytope.
