\begin{definition}[Definition:Bernoulli Numbers/Recurrence Relation]
The '''Bernoulli numbers''' $B_n$ are a sequence of rational numbers defined by the recurrence relation:
:$B_n = \begin{cases} 1 & : n = 0 \\ \ds - \sum_{k \mathop = 0}^{n - 1} \binom n k \frac {B_k} {n + 1 - k} & : n > 0 \end{cases}$
or equivalently:
:$B_n = \begin{cases} 1 & : n = 0 \\ \ds - \frac 1 {n + 1} \sum_{k \mathop = 0}^{n - 1} \binom {n + 1} k B_k & : n > 0 \end{cases}$
\end{definition}