Consider the following $n \times n$ matrices:
$$ A_{i,j} = \binom{2 m}{i - j + p},$$ $$ B_{i,j} = (i-j)^2 \binom{2 m}{i - j + p},$$
with $m \in \mathbb{N}^{+}$, $p \in \mathbb{N}$ and $p \leq 2 m$.
I am trying to show that $$\mathrm{tr}(A^{-1} B) = \frac{n(n-1)(p-2m)p}{2 m + 1}\;.$$
I can't find an explicit formula for $A^{-1}$, and I don't know if that is necessary. For what it's worth, with the help of https://arxiv.org/abs/math/9902004 one can show that $$ \det(A) = \prod_{j=1}^p \frac{(n+j)^{\overline{2 m - p}}}{j^{\overline{2 m - p}}},$$ where $n^{\overline{k}}$ denotes rising factorials.