I am trying to derive the formula $$\displaystyle\sum_{k=0}^{n}{2n\choose 2k}E_k = \displaystyle\sum_{k=0}^{n}{n\choose k}^2E_k=0$$
Where $E_k$ are the Euler Numbers.
The approach that I have taken is to first prove that ${2n\choose 2k} = {n\choose k}^2$. I looked into applying Pascal's Rule, but that got me nowhere.
I also have already proven that $E_{2n+1} = 0$ for $n=1,2,...$ I feel like this will come into play when I attempt to prove that the sequence is equal to $0$.