Sum of Euler Numbers by Binomial Coefficients Vanishes/Corollary

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Theorem

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

Then:

\(\displaystyle E_{2 n}\) \(=\) \(\displaystyle -\sum_{k \mathop = 0}^{n - 1} \dbinom {2 n} {2 k} E_{2 k}\)
\(\displaystyle \) \(=\) \(\displaystyle -\paren {\binom {2 n} 0 E_0 + \binom {2 n} 2 E_2 + \binom {2 n} 4 E_4 + \cdots + \binom {2 n} {2 n - 2} E_{2 n - 2} }\)

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:

\(\displaystyle \sum_{k \mathop = 0}^{n - 1} \dbinom {2 n} {2 k} E_{2 k} + E_{2 n}\) \(=\) \(\displaystyle 0\)
\(\displaystyle E_{2 n}\) \(=\) \(\displaystyle - \sum_{k \mathop = 0}^{n - 1} \dbinom {2 n} {2 k} E_{2 k}\)

$\blacksquare$