Definition:Euler Numbers/Recurrence Relation

Definition
The Euler numbers $E_n$ are a sequence of rational numbers defined by the recurrence relation:
 * $E_n = \begin{cases} 1 & : n = 0 \\ 0 & : n > 0, n = 2r + 1 \\ \displaystyle - \sum_{k \mathop = 0}^{n - 1} \binom {2n} {2k} E_{2k} & : n > 0, n = 2r \end{cases}$

Also see

 * Sum of Euler Numbers by Binomial Coefficients Vanishes