Algebraic representatives of the ratios $\zeta (2n+1)\mathrm{/}{\pi }^{2n}$ and $\beta (2n)\mathrm{/}{\pi }^{2n-1}$

In \cite{TallaWaffo2025arxiv2511.02843} we introduced even polynomials ${\mathrm{\Xi }}_n,{\mathrm{\Lambda }}_n\in \mathbb{Q}[x]$ arising from integral representations of $\beta (2n)\mathrm{/}{\pi }^{2n-1}$ and $\zeta (2n+1)\mathrm{/}{\pi }^{2n}$. In this paper we give explicit closed formulae for these polynomials in terms of Eulerian numbers and study their structural properties. These properties may prove useful in studies on the arithmetic nature of the ratios $\beta (2n)\mathrm{/}{\pi }^{2n}$ and $\zeta (2n+1)\mathrm{/}{\pi }^{2n+1}\mathrm{.}$