Consider the following $n \times n$ upper triangular matrix with a particularly nice structure:

\begin{equation}\mathbf{P} = \begin{pmatrix} 1 & \beta & \alpha+\beta & \dots & (n-3)\alpha + \beta & (n-2)\alpha + \beta\\ 0 & 1 & \beta & \dots & (n-4)\alpha + \beta & (n-3)\alpha + \beta\\ 0 & 0 & 1 & \dots & (n-5)\alpha + \beta & (n-4)\alpha + \beta\\ \vdots & \vdots & \vdots & \ddots & \vdots & \vdots \\ 0 & 0 & 0 & \dots & \beta & \alpha+\beta\\ 0 & 0 & 0 & \dots & 1 & \beta\\ 0 & 0 & 0 & \dots & 0 & 1 \end{pmatrix} \end{equation}

i.e.

\begin{equation} p_{i,j}=\begin{cases} 0, &i>j,\\ 1, &i=j,\\ (j-i-1)\alpha+\beta, &i<j. \end{cases} \end{equation}

Would it be possible to find an explicit expression for the elements of the inverse of $\mathbf{P}$?