Polynomial formula conjecture for the fixed-degree coefficients $H_n[d]$

Let $H_n[d]$ be the fixed-degree coefficients considered in the generating series, and for each $d$ define

$$\mathscr{H}(x;d)=\sum_{n\geq d+3}\frac{H_n[d]}{\pi^{2(n-3-d)}}\frac{x^{n-1}}{(n-1)!}.$$

For each $d\geq 0$, there exists a polynomial $P_d$ of degree $d$ with rational coefficients such that

$$H_n[d]=\frac{2d+1}{(2d-1)!!}\frac{P_d(n)}{2^{n-3-d}}\frac{(2(n-3-d))!}{(n-3-d)!}\pi^{2(n-3-d)}.$$

The formula for $d=0$ uses the convention $(-1)!!=1$. Equivalently, there exists $\widetilde{P}_d\in\mathbb{Q}[x]$ such that

$$\mathscr{H}(x;d)=\left[\widetilde{P}_d(x)(1-x)^{3/2}\right]_{\geq d+2},$$

where the truncation keeps only monomials of degree greater than $d+2$. Polynomial formula conjecture. For every $d\geq 0$, the displayed formula holds for the coefficients $H_n[d]$, equivalently with the stated generating-series representation.