Xin–Zhang's tridiagonal determinant conjecture

Let $n$ be a positive integer, let $t$ be a parameter, and let $C$ be the $(n-1)\times(n-1)$ matrix with entries

$$C_{i,j}=\begin{cases} t-(n-2i+2)(n-1-i)-1,&\text{if }i=j;\\ (i-1)(n-i),&\text{if }i=j+1;\\ -(n-1-i)(n-i),&\text{if }i=j-1;\\ 0,&\text{otherwise}.\end{cases}$$

Xin–Zhang's determinant conjecture. The determinant of $C$ is

$$\det C=\begin{cases}(t-n+1)\displaystyle\prod_{i=0}^{r-2}\bigl(t-n+1-r(r-1)+i(i+1)\bigr)^2,&\text{if }n=2r;\\ (t-n+1)\bigl(t-n+1-(r-1)^2\bigr)\displaystyle\prod_{i=1}^{r-2}\bigl(t-n+1-(r-1)^2+i^2\bigr)^2,&\text{if }n=2r-1.\end{cases}$$

This product formula gives the characteristic polynomial associated with a recurrence enumerating nonnegative integer matrices with equal row and column sums, and is related to the Ehrhart polynomial of the Birkhoff polytope. The paper proves the conjecture by showing that a scalar shift of $C$ is similar to a lower triangular matrix, so its characteristic polynomial factors through the diagonal entries.