  Through a brute-force approach to calculating the higher derivatives of the
falling factorial function, a number of interesting quantities were obtained
and analyzed. In particular, it was found that a quantity that can be described
as the "elementary symmetric harmonic sum" is a natural way to describe the
solutions to such higher derivatives. The relationship of this type of
generalized harmonic number to other quantities discussed in the literature,
such as the r-Stirling numbers, was described in detail.
