Let $n \in \mathbb { N }$. Deduce from the previous parts the value of the determinant $$H _ { n } = \operatorname { det } \left( C _ { i + j - 2 } \right) _ { 1 \leqslant i , j \leqslant n + 1 } = \left| \begin{array} { c c c c c c } C _ { 0 } & C _ { 1 } & C _ { 2 } & \ldots & C _ { n - 1 } & C _ { n } \\ C _ { 1 } & \therefore & & \therefore & \therefore & C _ { n + 1 } \\ C _ { 2 } & & \ddots & \ddots & \therefore & \vdots \\ \vdots & \therefore & \ddots & \ddots & & C _ { 2 n - 2 } \\ C _ { n - 1 } & \therefore & \ddots & & \therefore & C _ { 2 n - 1 } \\ C _ { n } & C _ { n + 1 } & \cdots & C _ { 2 n - 2 } & C _ { 2 n - 1 } & C _ { 2 n } \end{array} \right|.$$
Let $n \in \mathbb { N }$. Deduce from the previous parts the value of the determinant
$$H _ { n } = \operatorname { det } \left( C _ { i + j - 2 } \right) _ { 1 \leqslant i , j \leqslant n + 1 } = \left| \begin{array} { c c c c c c } C _ { 0 } & C _ { 1 } & C _ { 2 } & \ldots & C _ { n - 1 } & C _ { n } \\ C _ { 1 } & \therefore & & \therefore & \therefore & C _ { n + 1 } \\ C _ { 2 } & & \ddots & \ddots & \therefore & \vdots \\ \vdots & \therefore & \ddots & \ddots & & C _ { 2 n - 2 } \\ C _ { n - 1 } & \therefore & \ddots & & \therefore & C _ { 2 n - 1 } \\ C _ { n } & C _ { n + 1 } & \cdots & C _ { 2 n - 2 } & C _ { 2 n - 1 } & C _ { 2 n } \end{array} \right|.$$