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:


 * $\begin{bmatrix} b_{i j} \end{bmatrix} = \begin{bmatrix} \dfrac {\displaystyle \prod_{k \mathop = 1}^n \left({i + k - 1}\right) \left({j + k - 1}\right)} {\displaystyle \left({i + j - 1}\right) \left({\prod_{\substack {1 \mathop \le k \mathop \le n \\ k \mathop \ne i} } \left({i - k}\right)}\right) \left({\prod_{\substack {1 \mathop \le k \mathop \le n \\ k \mathop \ne j} } \left({j - k}\right)}\right)} \end{bmatrix}$

We address in turn the various expressions that go to form this.