The polynomial form of $S_{k,j}(n)$

Let $t:=n(n+1)$, and define

$$g_1(n):=1,\qquad g_j(n):=\frac{j}{(j-1)!}\prod_{q=1}^{j-1}(jn+q)\quad (j\geq 2).$$

Polynomial-factorization conjecture. There exist polynomials $P_k(t,x)$ in $x$ of degree $k$, with coefficients rationally depending on $t$, such that

$$S_{k,j}(n)=\frac{t^{k+1}}{2^{k+1}}\left.P_k(t,x)\right|_{x=j}g_j(n).$$

The conjecture is the paper's main conjecture for the general sums; the source records explicit examples through $k=7$ and notes that no recursion relation is known in general, although specialization at $t=2$ recovers Tuenter's polynomials.