Equivalence of Definitions of Bernoulli Numbers

Proof
From the generating function definition:

Equating coefficients:

For $n = 0$:

For $n > 0$:

Hence the result:
 * $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}$