For $n \in \mathbb{N}^*$ and $(i,j) \in \llbracket 1, n \rrbracket^2$, we denote by $h_{i,j}^{(-1,n)}$ the coefficient at position $(i,j)$ of the matrix $H_n^{-1}$ and we denote by $s_n$ the sum of the coefficients of the matrix $H_n^{-1}$, that is: $$s_n = \sum_{1 \leqslant i,j \leqslant n} h_{i,j}^{(-1,n)}$$ Let $n \in \mathbb{N}^*$. a) Show that there exists a unique $n$-tuple of real numbers $\left(a_p^{(n)}\right)_{0 \leqslant p \leqslant n-1}$ satisfying the following system of $n$ linear equations in $n$ unknowns: $$\left\{\begin{array}{ccccccc} a_0^{(n)} + & \frac{a_1^{(n)}}{2} + \cdots + \frac{a_{n-1}^{(n)}}{n} = & 1 \\ \frac{a_0^{(n)}}{2} + \frac{a_1^{(n)}}{3} + \cdots + \frac{a_{n-1}^{(n)}}{n+1} = & 1 \\ \vdots & \vdots & & \vdots \\ \frac{a_0^{(n)}}{n} + \frac{a_1^{(n)}}{n+1} + \cdots + \frac{a_{n-1}^{(n)}}{2n-1} = & 1 \end{array}\right.$$ b) Show that $s_n = \sum_{p=0}^{n-1} a_p^{(n)}$.
For $n \in \mathbb{N}^*$ and $(i,j) \in \llbracket 1, n \rrbracket^2$, we denote by $h_{i,j}^{(-1,n)}$ the coefficient at position $(i,j)$ of the matrix $H_n^{-1}$ and we denote by $s_n$ the sum of the coefficients of the matrix $H_n^{-1}$, that is:
$$s_n = \sum_{1 \leqslant i,j \leqslant n} h_{i,j}^{(-1,n)}$$
Let $n \in \mathbb{N}^*$.
a) Show that there exists a unique $n$-tuple of real numbers $\left(a_p^{(n)}\right)_{0 \leqslant p \leqslant n-1}$ satisfying the following system of $n$ linear equations in $n$ unknowns:
$$\left\{\begin{array}{ccccccc} a_0^{(n)} + & \frac{a_1^{(n)}}{2} + \cdots + \frac{a_{n-1}^{(n)}}{n} = & 1 \\ \frac{a_0^{(n)}}{2} + \frac{a_1^{(n)}}{3} + \cdots + \frac{a_{n-1}^{(n)}}{n+1} = & 1 \\ \vdots & \vdots & & \vdots \\ \frac{a_0^{(n)}}{n} + \frac{a_1^{(n)}}{n+1} + \cdots + \frac{a_{n-1}^{(n)}}{2n-1} = & 1 \end{array}\right.$$
b) Show that $s_n = \sum_{p=0}^{n-1} a_p^{(n)}$.