Hint on power sum coefficients Please do not give anything more than a tiny hint for this question.
I know that there is a well-known formula for $$\sum_{i=1}^n i^k,$$ where $k$ is any non-negative integer. I have been able to prove that in fact it is a polynomial in $n$,
$$\sum_{i=1}^n i^k = \sum_{j=0}^{k+1} a_j n^j,$$
with high-order term $\frac 1 {k+1} n^{k+1}$ and zero constant term. In the process, I found a rather awkward method of calculating the rest of the coefficients. I'm now trying to figure out what the rest of them are. So far, I've gotten
$$a_j = \frac{k!}{j!(k-j+1)!}-\sum_{m=j+1}^{k+1}a_{m}\frac{m!}{j!(m-j+1)!},$$
where $a_j$ is the coefficient of $n^j$ (for $j\le k$). Can someone give me a tiny hint on how to proceed? Please do not go and tell me what the coefficients are, or how the rest of the proof goes, or anything like that.

Look at the Riemann and Hurwitz zeta functions.
