From Hilbert Matrix is Cauchy Matrix, $H_n$ is a special case of a Cauchy matrix: :$\begin{bmatrix} c_{i j} \end{bmatrix} = \begin{bmatrix} \dfrac 1 {x_i + y_j} \end{bmatrix}$ where: :$x_i = i$ :$y_j = j - 1$ From Inverse of Cauchy Matrix, the inverse of the square Cauchy matrix of order $n$ is: :$\begin{bmatrix} b_{i j} \end{bmatrix} = \begin{bmatrix} \dfrac {\displaystyle \prod_{k \mathop = 1}^n \paren {x_j + y_k} \paren {x_k + y_i} } {\displaystyle \paren {x_j + y_i} \paren {\prod_{\substack {1 \mathop \le k \mathop \le n \\ k \mathop \ne j} } \paren {x_j - x_k} } \paren {\prod_{\substack {1 \mathop \le k \mathop \le n \\ k \mathop \ne i} } \paren {y_i - x_k} } } \end{bmatrix}$ Thus $H_n^{-1}$ can be specified as: :$\begin{bmatrix} b_{i j} \end{bmatrix} = \begin{bmatrix} \dfrac {\displaystyle \prod_{k \mathop = 1}^n \paren {i + k - 1} \paren {j + k - 1} } {\displaystyle \paren {i + j - 1} \paren {\prod_{\substack {1 \mathop \le k \mathop \le n \\ k \mathop \ne i} } \paren {i - k} } \paren {\prod_{\substack {1 \mathop \le k \mathop \le n \\ k \mathop \ne j} } \paren {j - k} } } \end{bmatrix}$ First, from Product of Products: :$\displaystyle \prod_{k \mathop = 1}^n \paren {i + k - 1} \paren {j + k - 1} = \prod_{k \mathop = 1}^n \paren {i + k - 1} \prod_{k \mathop = 1}^n \paren {j + k - 1}$ We address in turn the various factors of this expression for $b_{i j}$. {{begin-eqn}} {{eqn | l = \prod_{k \mathop = 1}^n \paren {i + k - 1}       | r = \prod_{k \mathop = 0}^{n - 1} \paren {i + k}       | c = Translation of Index Variable of Product }} {{eqn | r = i^{\overline n}       | c = Definition of Rising Factorial }} {{eqn | r = \frac {\paren {i + n - 1}!} {\paren {i - 1}!}       | c = Rising Factorial as Quotient of Factorials }} {{end-eqn}} and similarly: :$\displaystyle \prod_{k \mathop = 1}^n \paren {j + k - 1} = \frac {\paren {j + n - 1}!} {\paren {j - 1}!}$ Then: {{begin-eqn}} {{eqn | l = \prod_{\substack {1 \mathop \le k \mathop \le n \\ k \mathop \ne i} } \paren {i - k}       | r = \paren {\prod_{1 \mathop \le k \mathop < i} \paren {i - k} } \paren {\prod_{i \mathop < k \mathop \le n} \paren {i - k} }       | c =  }} {{eqn | r = \paren {i - 1}! \paren {\prod_{i \mathop < k \mathop \le n} \paren {i - k} }       | c = {{Defof|Factorial}} }} {{eqn | r = \paren {i - 1}! \paren {\prod_{0 \mathop < k \mathop \le n - i} \paren {-k} }       | c = Translation of Index Variable of Product }} {{eqn | r = \paren {i - 1}! \paren {-1}^{n - i} \paren {\prod_{0 \mathop < k \mathop \le n - i} k}       | c =  }} {{eqn | r = \paren {i - 1}! \paren {-1}^{n - i} \paren {n - i}!       | c = {{Defof|Factorial}} }} {{end-eqn}} and similarly: :$\displaystyle \prod_{\substack {1 \mathop \le k \mathop \le n \\ k \mathop \ne j} } \paren {j - k} = \paren {j - 1}! \paren {-1}^{n - j} \paren {n - j}!$ Thus we can write: {{begin-eqn}} {{eqn | l = \begin{bmatrix} b_{i j} \end{bmatrix}       | r = \frac {\paren {\dfrac {\paren {i + n - 1}!} {\paren {i - 1}!} } \paren {\dfrac {\paren {j + n - 1}!} {\paren {j - 1}!} } } {\paren {i + j - 1} \paren {i - 1}! \paren {-1}^{n - i} \paren {n - i}! \paren {j - 1}! \paren {-1}^{n - j} \paren {n - j}!}       | c =  }} {{eqn | r = \frac {\paren {-1}^{i + j} \paren {i + n - 1}! \paren {j + n - 1}!} {\paren {\paren {i - 1}!}^2 \paren {\paren {j - 1}!}^2 \paren {n - i}! \paren {n - j}! \paren {i + j - 1} }       | c =  }} {{end-eqn}} {{qed}}