Asymptotic ratios of Bernoulli partition terms

Prove that, for every fixed column index n greater than or equal to 2, the asymptotic ratio of the Bernoulli-partition entry b_m(n) to the absolute Bernoulli number |B_{2m}| satisfies lim_{m\to\infty} b_m(n)/|B_{2m}| = p_n(\pi^2), where p_n(x) is the polynomial defined by p_n(x) = ((4n^2-1))\sum_{k=0}^{\lfloor n/2\rfloor-1} [(-4)^k x^{k+1}/(2k+1)!] [\Gamma(n-1)\Gamma(2n-2-2k)/(\Gamma(n-1-2k)\Gamma(2n-2))].

Background

The paper expresses each absolute even-index Bernoulli number |B_{2m}| as a finite sum of positive, monotonically falling rational numbers b_m(n), obtained from the inverse of a triangular matrix derived from finite-sum identities for Bernoulli numbers. For each fixed column index n, numerical calculations indicate that b_m(n)/|B_{2m}| approaches a limit a(n) as the row index m tends to infinity.

The conjecture identifies these limiting ratios with the values p_n(\pi^2) of an explicitly defined family of polynomials. Establishing this formula would rigorously explain the observed asymptotic proportions of the Bernoulli partitions and, together with the claimed normalization of the ratios, characterize the limiting infinite partition associated with the Bernoulli numbers.