(I have no idea why this was closed, nor did I receive any information/email regarding what details or clarity were missing. I have rendered the Mathematica into mathematics.)
copied from mathematica.stackexchange.com
Here I asked about a symbolic summation, 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 summation, which in a general format is:
$$\sum _{p=2}^m (p-1)^s p^t$$
with the assumptions that $(s|t)\in \mathbb{N}$. 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:
$$\sum _{p=2}^m (p-1)^{i-k-1} p^{n-i}$$
with integers $m \ge 1$ and $n \ge 2$, $i$ 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.