A problem in theoretical physics has led me to consider a Hankel matrix $H$ whose elements are given by $H_{ij} = (i+j-1)^{i+j-1}$, where $i, j = \left[1, N\right]$ for any $N$ you like. I'm curious to know if there is any hope of inverting this matrix, ideally in such a way that we can learn properties of the inverse in the large $N$ limit.
Of course, we can make progress numerically for sufficiently small $N$, but these matrix elements grow rapidly and it isn't long before machine precision is an issue. It seems that an analytical insight is necessary, but I'm not familiar with results for Hankel/Toeplitz matrices and have no idea if a solution for this problem is within reach. Any literature recommendations for this subject are welcome, in addition to any examples which can provide intuition for how the Hankel matrix structure can help in finding the inverse.