Definition:Bernoulli Numbers

Definition
The Bernoulli numbers $B_n$ are a sequence of rational numbers defined by:

Generating Function
The exponential generating function:


 * $\displaystyle \frac x {e^x - 1} = \sum_{n \mathop = 0}^\infty \frac{B_n x^n} {n!}$

Recurrence Relation
The recurrence relation:


 * $B_0 = 1; \ \displaystyle B_n = - \sum_{k \mathop = 0}^{n-1} \binom {n} {k} \frac {B_k} {n - k + 1}$

The values of the first Bernoulli Numbers are:


 * $1, - \dfrac 1 2, \dfrac 1 6, 0, - \dfrac 1 {30}, 0, \dfrac 1 {42}, 0, - \dfrac 1 {30} \ldots$

Also see

 * Equivalence of Definitions of Bernoulli Numbers