Let $v(x) = \frac{1}{\cos(x)}$ on $]-\frac{\pi}{2}, \frac{\pi}{2}[$ and $E_{2k} = v^{(2k)}(0)$ for $k \in \mathbb{N}$.
Show that, for $n \in \mathbb{N}^*$,
$$\sum_{k=0}^{n} (-1)^k \binom{2n}{2k} E_{2k} = 0$$
and deduce the values of $E_0$, $E_2$ and $E_4$.