It's strange where things can lead. I wondered if a matrix of consecutive integers had a special name in standard mathematical usage. I couldn't find such a name, but I did find a few with special names, including one called the Lehmer Matrix. Before I knew what was happening, I'd created a Mathcad worksheet implementing the Lehrer Matrix and its inverse, which has an explicit formula.
The worksheet is attached. I can't check it properly because I'm about to run out of my 30-day trial period and lose both symbolics and programming.
Please attach also a PDF file to preview the contents of your worksheet.
As requested, Vladimir, plus (very slightly) amended worksheet ... please ignore the last page, I was just playing.
Oh, and a link to the Wiki page on the Lehmer Matrix: https://en.wikipedia.org/wiki/Lehmer_matrix