This is a Hilbert matrix. There is an outline proof that its inverse has integer entries in the delightful paper Tricks or treats with the Hilbert matrix, by Man-Duen Choi, available as a pdf download here.
This is a Hilbert matrix. There is an outline proof that its inverse has integer entries in the delightful paper Tricks or treats with the Hilbert matrix, by Man-Duen Choi, available as a pdf download here.