Determinant of a certain Toeplitz matrix 
Compute the following determinant
$$\begin{vmatrix} x & 1 & 2 & 3 & \cdots & n-1 & n\\ 1 & x & 1 & 2 & \cdots & n-2 & n-1\\ 2 & 1 & x & 1 & \cdots & n-3 & n-2\\ 3 & 2 & 1 & x & \cdots & n-4 & n-3\\ \vdots & \vdots & \vdots & \vdots & \ddots & \vdots & \vdots\\ n-1 & n-2 & n-3 & n-4 & \cdots & x & 1\\ n & n-1 & n-2 & n-3 & \cdots & 1 &x \end{vmatrix}$$


I tried the following. I subtracted the second row from the first, the third from the second, the fourth from the third, and so on. I got:
\begin{vmatrix} x-1 & 1-x & 1 & 1 & \cdots & 1 & 1\\ -1 & x-1 & 1-x & 1 & \cdots & 1 & 1\\ -1 & -1 & x-1 & 1-x & \cdots & 1 & 1\\ 3 & 2 & 1 & x & \cdots & n-4 & n-3\\ \vdots & \vdots & \vdots & \vdots & \ddots & \vdots & \vdots\\ -1 & -1 & -1 & -1 & \cdots & x-1 & 1-x\\ n & n-1 & n-2 & n-3 & \cdots & 1 &x \end{vmatrix}
I did the same thing with the columns. I subtracted the second row from the first, the third from the second, the fourth from the third, and so on. And I got:
\begin{vmatrix} 2x-2 & -x & 0 & 1 & \cdots & 0 & 1\\ -x & 2x-2 & -x & 1 & \cdots & 0 & 1\\ -2 & -x & 2x-2 & 1-x & \cdots & 0 & 1\\ 1 & 1 & 1-x & x & \cdots & -1 & n-3\\ \vdots & \vdots & \vdots & \vdots & \ddots & \vdots & \vdots\\ -2 & -2 & -2 & -1 & \cdots & 2x-2 & 1-x\\ 1 & 1 & 1 & n-3 & \cdots & 1-x &x \end{vmatrix}
I hope I didn’t make a mistake somewhere. With this part I don't know what to do next. I don't know if I'm doing it right. Thank you in advance !

Here are the first few determinants with the help of WA:
$$
\begin{array}{rl}
n & \text{determinant}
\\ 1 & x^2 - 1
\\ 2 & x^3 - 6 x + 4
\\ 3 & x^4 - 20 x^2 + 32 x - 12
\\ 4 & x^5 - 50 x^3 + 140 x^2 - 120 x + 32
\\ 5 & x^6 - 105 x^4 + 448 x^3 - 648 x^2 + 384 x - 80
\\ 6 & x^7 - 196 x^5 + 1176 x^4 - 2520 x^3 + 2464 x^2 - 1120 x + 192
\\ 7 & x^8 - 336 x^6 + 2688 x^5 - 7920 x^4 + 11264 x^3 - 8320 x^2 + 3072 x - 448
\end{array}
$$
There are some patterns for the coefficients but I don't see a complete pattern:

*

*The polynomial has degree is $n+1$


*The coefficient of $x^{n+1}$ is $1$


*The coefficient of $x^{n}$ is $0$


*The coefficient of $x^{n-1}$  is $-$A002415$(n+1)$


*The independent term is $(-1)^n$A001787$(n)$
OEIS doesn't have the sequence of coefficients of $x^{n-2}$ nor of $x$.
I don't expect a nice closed form in monomial form. A recurrence is more probable.
