For $p\in\mathbb{N}$ a monic polynomial $q_p$ exists with $S_p(n):=\sum_{k=1}^nk^p=\frac{n(n+1)}{p+1}q_p(n)$. After $q_1=1$ we seem to have an alternating pattern, with $$q_2=n+\frac{1}{2},\\q_3=n(n+1),\\q_4=(n^2+n-\frac{1}{3})q_2,\\q_5=(n^2+n-\frac{1}{2})q_3,\\q_6=(n^4+2n^3-n+\frac{1}{3})q_2,\\q_7=(n^4+2n^3-\frac{1}{3}n^2=\frac{4}{3}n+\frac{2}{3})q_3,\\q_8=(n^6+3n^5+n^4-3n^3-\frac{1}{5}n^5+\frac{9}{5}n-\frac{3}{5})q_2,\\q_9=(n^2+n-1)(n^4+2n^3-\frac{1}{2}n^2-\frac{3}{2}n+\frac{3}{2})q_3.$$In particular we have the following conjectures:
- For $k\ge 1$ degree-($2k-2$) monic polynomials $A_k,\,B_k$ exist for which $q_{2k}=q_2A_k,\,q_{2k+1}=q_3B_k$ so that $S_{2k}$ has repeated roots $-1,\,0$ and $S_{2k+1}$ has roots including $-1,\,-\frac{1}{2},\,0$;
- For $k\ge 2$ the polynomials $A_k,\,B_k$ have the same next-to-leading coefficients.
Can we prove, or at least make some sense of, any of these results? It feels like the $-\frac{1}{2}$ root and the repeated roots have some interpretation, and the fact that $A_k-B_k$ is of degree $\le 2k-4$, should have some significance.