The polynomial formula for asymptotic ratios of Bernoulli partitions

Let $b_m(n)$ denote the Bernoulli partition entries described in the preceding discussion, and let $B_{2m}$ be the Bernoulli numbers. For each column index $n\geq 2$, define the asymptotic ratio $a(n)$ by

$$\frac{b_m(n)}{|B_{2m}|}\underset{m\rightarrow\infty}{\sim}a(n).$$

Let $\lfloor\cdot\rfloor$ denote the floor function, and let $\Gamma$ denote the gamma function. Polynomial formula for the asymptotic ratios. The asymptotic ratios are $a(n)=p_n(\pi^2)$ for $n\geq2$, where $p_n(x)$ is the polynomial

$$p_n(x)=\frac{1}{4n^2-1}\sum_{k=0}^{\left\lfloor n/2\right\rfloor-1}\frac{(-4)^k x^{k+1}}{(2k+1)!}\frac{\Gamma(n-1)\Gamma(2n-2-2k)}{\Gamma(n-1-2k)\Gamma(2n-2)}.$$

The formula is significant because it gives an explicit expression for the limiting growth ratios of the Bernoulli partition entries in terms of polynomials evaluated at $\pi^2$; the source presents it as a consequence of numerical calculations and OEIS data, with no resolution evidence supplied.