For $n\in\mathbb{N}$ and $m=\lfloor\frac{n}2\rfloor$, consider the $n\times n$ skew-symmetric matrix $A_n$ where each entry in the first $m$ sub-diagonals below the main diagonal is $1$ and each of the remaining entries below the main diagonal is $-1$. Let $I_n$ be the $n\times n$ identity matrix.

Next, construct the matrix $M_n=A_n+xI_n$. For example, we have $$M_3=\begin{pmatrix} x&-1&1 \\ 1&x&-1 \\ -1&1&x \end{pmatrix} \qquad \text{and} \qquad M_4=\begin{pmatrix} x&-1&-1&1 \\ 1&x&-1&-1 \\ 1&1&x&-1 \\ -1&1&1&x \end{pmatrix}.$$

QUESTION.Is the following true? Experiments suggest to be so. $$\det(M_n)=\sum_{k=0}^m\binom{n}{2k}x^{n-2k}.$$