Given an $m \times m$ matrix $M$ with $i,j$-th entry $$ M_{i, j} = \frac{\frac{1}{i+1}+\frac{1}{j+1}}{i+j}, \quad i,j=1,\ldots,m $$
This is a Hilbert-like matrix. I have numerically checked that it is invertible but I don't know how to prove it.
I was thinking that $\frac{\frac{1}{i+1}+\frac{1}{j+1}}{i+j} = \frac{1}{(1+i)(1+j)} + \frac{2}{(1+i)(1+j)(i+j)}$, so to prove that $M$ is invertible, it suffices to show that another matrix $N$ with entry $[N]_{i,j} = \frac{1}{(1+i)(1+j)(i+j)}$ is invertible.
The matrix $N$ is invertible as the product of invertible matrices $DHD,$ where $M$ is the Hilbert matrix and $D$ is the diagonal matrix with entries $(1+i)^{-1}.$
The sum is invertible as $DMD$ and the one dimensional matrix with entries $(1+i)^{-1}(1+j)^{-1}$ are positive definite and $DMD$ is invertible.