Definition:Bernoulli Numbers/Recurrence Relation

Definition
The Bernoulli numbers $B_n$ are a sequence of rational numbers defined by the recurrence relation:
 * $B_n = \begin{cases} 1 & : n = 0 \\ \displaystyle - \sum_{k \mathop = 0}^{n - 1} \binom n k \frac {B_k} {n - k + 1} & : n > 0 \end{cases}$

Illustration


 * $\begin{array}{r|cccccccccc}

B_N & \binom n 0 & \binom n 1 & \binom n 2 & \binom n 3 & \binom n 4 & \binom n 5 & \cdots \\ \hline

B_0 = 1 & 1 B_0  &  &  &  &  &  & \cdots & = 1\\

B_1 = - \frac {1} {2} & 1 B_0  & +2 B_1 &  &  &  &  & \cdots & = 0\\

B_2 = + \frac {1} {6} & 1 B_0  & +3 B_1 & + 3 B_2 &  &  &  & \cdots & = 0\\

B_3 = 0 & 1 B_0  & +4 B_1 & +6 B_2 & + 4 B_3 &  &  & \cdots & = 0\\

B_4 = - \frac {1} {30} & 1 B_0  & +5 B_1 & +10 B_2 & +10 B_3 & +5 B_4 &  & \cdots & = 0\\

B_5 = 0 & 1 B_0  & +6 B_1 & +15 B_2 & +20 B_3 & +15 B_4 & +6 B_5 & \cdots & = 0\\

\cdots & \cdots & \cdots & \cdots & \cdots & \cdots & \cdots & \cdots & = 0 \\

\end{array}$

Also see

 * Equivalence of Definitions of Bernoulli Numbers