The Bernoulli-number formula for the terminal coefficients $p_{k,k-1}(n)$

Let $t:=n(n+1)$, and define $p_{k,k-1}(n):=p_{k,k-1}(t)|_{t=n(n+1)}$. Suppose

$$p_{k,k-1}(n)=c_k\frac{v_k(n)}{n^{k-1}(n+1)^{k-1}},$$

where $v_k(n)$ is monic of degree $2k-2$. Let $B_{2k}$ denote the Bernoulli numbers, defined by

$$\frac{x}{e^x-1}=\sum_{r\geq 0}B_r\frac{x^r}{r!}.$$

Terminal-coefficient conjecture. The polynomials and constants are given by

$$v_k(n)=\sum_{q=1}^{2k-1}\frac{{2k\choose q}+(-1)^{q+1}}{2k+1}n^{2k-q-1},$$

and

$$c_k=(-1)^{k+1}(2k+1)2^kB_{2k}.$$

The formulas are conjectured from the displayed cases and are presented in the section on the conjectural relationship with simple symmetric Venn diagrams; the source gives no proof or resolution.