For $n \in \mathbb{N}$ and $k \in \llbracket 0; n \rrbracket$, we denote by $\binom{n}{k}$ the binomial coefficient $\binom{n}{k} = \frac{n!}{k!(n-k)!}$. The family $\left(K_n\right)_{n \in \mathbb{N}}$ is the orthonormal family defined in question II.E.
For all $n \in \mathbb{N}$, show that we can write:
$$K_n = \sqrt{2n+1} \, \Lambda_n$$
where $\Lambda_n$ is a polynomial with integer coefficients that we will make explicit.
Among the coefficients of $\Lambda_n$, which ones are even?