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As observed in comments, the problem is equivalent to finding the inverse of the matrices $$H_\lambda:=\Big[{1\over i+j-\lambda}\Big]_{ij},$$ for $\lambda=1/2$ and $\lambda=3/2$. These are also particular cases of Cauchy matrices, which admit an explicit formula for the inverse matrix (see the wiki article on Cauchy matrix and Fedor Petrov's answer in the linked question).