I'm trying to show that $\sum_{i = 0}^{p-2} (i+1)^{-1} t^{i+n}$ where $1 \leq n \leq p-2$ spans the vector space $\mathbb{F}_p[t]/(1-t)^{p-1}$ as a rank $p-1$ module over $\mathbb{F}_p$.

In other words, I would like to show that the determinant of the following matrix is a unit in $\mathbb{F}_p$. I've shown this for $p = 2, 3, 5, 7, 11, 13$. I have tried to use induction but failed. This is such a natural matrix I am hoping someone recognizes it!

$$\begin{pmatrix} 1 & 0 & -1 & (p-2)^{-1} & \cdots & 4^{-1} & 3^{-1} \\ 2^{-1} & 1 & 0 & -1 & (p-2)^{-1} & \cdots & 4^{-1} \\ 3^{-1} & 2^{-1} & 1 & 0 & -1 & \ddots & \vdots \\ \vdots & 3^{-1} & 2^{-1} & 1 & 0 & \ddots & (p-2)^{-1} \\ (p-3)^{-1} & \ddots & \ddots & \ddots & \ddots & 0 & -1 \\ (p-2)^{-1} & (p-3)^{-1} & (p-4)^{-1} & \ddots & 2^{-1} & 1 & 0 \\ -1 & (p-2)^{-1} & (p-3)^{-1} & \cdots & 3^{-1} & 2^{-1} & 1 \end{pmatrix}$$