How to invert a symmetric Toeplitz matrix?

Question

Consider the following equation, for $1\leq i\leq n$, $$ y_i=\sum_{k=0}^n \sum_{j=\max(-k,1-i)}^{\min(k,n-i)} x_{i+j} $$ which, for each $i$ and $k$, gives the sum of the terms within an index radius $k$. The $\max$ and $\min$ conditions exclude indexes outside $[1,n]$ For example, with $n=4$ and $i=2$, $$ \begin{aligned} y_2=\sum_{k=0}^4 \sum_{j=\max(-k,-1)}^{\min(k,2)} x_{i+j}=4x_1 +5x_2+4x_3+2x_4 \end{aligned} $$ Hence, we may alternatively write $$ y_i = \sum_{j=1}^n (n+1-|i-j|)x_j $$ In matrix form, with $\mathbf{y}=(y_1,...,y_n)^T$ and $\mathbf{x}=(x_1,...,x_n)^T$, we have $$ \mathbf{y}= A_n\mathbf{x} $$ where $$ A_n= \begin{pmatrix} n+1 & n & n-1 & n-2 & \cdots & 2\\ n & n+1 & n & n-1 & \cdots & 3\\ n-1 & n & n+1 & n & \cdots & 4\\ \vdots & & & & \ddots & \\ 2 & 3 & 4 & 5 & \cdots & n+1\\ \end{pmatrix} $$ For which $n$ is $A_n$ invertible?