Elements of Inverse of Hilbert Matrix are Integers

Theorem
Let $H_n$ be the Hilbert matrix of order $n$:


 * $\begin{bmatrix} a_{i j} \end{bmatrix} = \begin{bmatrix} \dfrac 1 {i + j - 1} \end{bmatrix}$

Consider its inverse $H_n^{-1}$.

All the elements of $H_n^{-1}$ are integers.

Proof
From Inverse of Hilbert Matrix, $H_n^{-1} = \left[{b}\right]_n$ can be specified as:


 * $\begin{bmatrix} b_{i j} \end{bmatrix} = \begin{bmatrix} \dfrac {\left({-1}\right)^{i + j} \left({i + n - 1}\right)! \left({j + n - 1}\right)!} {\left({\left({i - 1}\right)!}\right)^2 \left({\left({j - 1}\right)!}\right)^2 \left({n - i}\right)! \left({n - j}\right)! \left({i + j - 1}\right)} \end{bmatrix}$

Thus: