Equivalence of Definitions of Bernoulli Numbers

Theorem
The following definitions of the Bernoulli Numbers are equivalent:

Proof
Starting with the Generating Function definition:

By equating coefficients:


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

Solving this 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}$