Here I asked the question below, of the Mathematica community. I would be just as happy to solve it other ways -- even analytically! :-)
Here I asked about a symbolic sum, and received three very insightful replies (from: მამუკა ჯიბლაძე, Carl Woll and Dr. Wolfgang Hintze) which did the trick. (Thank you again!)
Currently I am trying to solve a different symbolic sum, which in a general format is:
Sum[(-1 + p)^s p^t, {p, 2, m}, Assumptions -> (s | t) \[Element] Integers]
Each factor alone would lead to Harmonic numbers or Zeta functions, respectively. However, in this form, the sum returns unevaluated. I tried all of the three methods suggested previously (including integrating instead of differentiating, in one of the approaches), to no avail.
More specifically, the sum is:
Sum[(-1 + p)^(-1 + i - k) p^(-i + n), {p, 2, m}]
with m and n any positive integers, i values running from 1 to n-1 and k from 0 to i-1.
I also tried replacing the (p-1)^s with its own sum (with the binomial coefficient, etc.) and switching the order of the summations -- also to no avail.
Thank you for any direction, and for your time.