Let $(H_{n})_{n \in \mathbb{N}}$ be the sequence of polynomials defined by: $\forall n \in \mathbb{N},\, H_{n}(X) = (-1)^{n} B_{n}(1-X)$, where $(B_n)_{n\in\mathbb{N}}$ are the Bernoulli polynomials satisfying
$$\begin{cases} B_{0} = 1 \\ \forall n \in \mathbb{N}^{*},\, B_{n}' = n B_{n-1} \\ \forall n \in \mathbb{N}^{*},\, \displaystyle\int_{0}^{1} B_{n}(t)\,\mathrm{d}t = 0 \end{cases}$$
Show that for all $n \in \mathbb{N}$, $H_{n} = B_{n}$.