\begin{definition}[Definition:Euler Numbers/Recurrence Relation]
The '''Euler numbers''' $E_n$ are a sequence of rational numbers defined by the recurrence relation:
:$E_n = \begin {cases} 1 & : n = 0 \\ 0 & : n > 0, n = 2 r + 1 \\ \ds -\sum_{k \mathop = 0}^{n - 1} \binom {2 n} {2 k} E_{2 k} & : n > 0, n = 2 r \end {cases}$
\end{definition}