Definition:Bernoulli Numbers/Archaic Form/Definition 1

Definition
The old form Bernoulli numbers ${B_n}^*$ are a sequence of rational numbers defined by the exponential generating function:
 * $\displaystyle \frac x {e^x - 1} = 1 - \frac x 2 + \sum_{n \mathop = 1}^\infty \left({-1}\right)^{n - 1} \frac{B_n^* x^{2 n} } {\left({2 n}\right)!}$

for $x \in \R$ such that $\left\lvert{x}\right\rvert < 2 \pi$

Also see

 * Equivalence of Definitions of Archaic Form of Bernoulli Numbers