# Sum of Euler Numbers by Binomial Coefficient

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## Theorem

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

Then:

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

\(\displaystyle \) | \(=\) | \(\displaystyle \binom {2 n} 2 E_{2 n - 2} + \binom {2 n} 4 E_{2 n - 4} + \binom {2 n} 6 E_{2 n - 6} + \cdots + 1\) |

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

## Proof

## Sources

- 1968: Murray R. Spiegel:
*Mathematical Handbook of Formulas and Tables*... (previous) ... (next): $\S 21$: Relationships of Bernoulli and Euler Numbers: $21.6$