Sum of Euler Numbers by Binomial Coefficients Vanishes/Corollary

Theorem
Let $n \in \Z_{>0}$ be a (strictly) positive integer.

Then:

where $E_n$ denotes the $n$th Euler number.

Proof
From Sum of Euler Numbers by Binomial Coefficients Vanishes we have:

$\forall n \in \Z_{>0}: \displaystyle \sum_{k \mathop = 0}^n \binom {2 n} {2 k} E_{2 k} = 0$

If:

$\forall n \in \Z_{>0}: \displaystyle \sum_{k \mathop = 0}^n \binom {2 n} {2 k} E_{2 k} = 0$

Then: