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Let $a(n,k)=(-1)^k {{2n-k}\choose k}$ for $0 \le k \le n$ and $a(n,k)=0$ else. Then it is known (cf. OEIS A005439 and A098435) that the first column of the inverse matrix of $(a(i,j))_{i,j\ge0}$ is ...

Let $n \geq 2$. Let $M_n$ be the $(n+1) \times (n+1)$ matrix with entries $\binom{l}{k}$ for $0 \leq l,k \leq n$ and $U_n=M_n + M_n^T$ and let $f_n(x)$ denote the characteristic polynomial of $U_n$. ...