Bernoulli number

Bernoulli numbers were defined by Jacob Bernoulli in connection with evaluating sums of the form ∑ $i^{k}$.
The sequence $B_{0} , B_{1} , B_{2} , ...$ can be generated using the formula
$x/(e^{x} - 1) = \sum (B_{n}x^{n})/n!$
though various different notations are used for them.
The first few are: $B_{0} = 1 , B_{1} = -{{1}\over{2}} , B_{2} = {{1}\over{6}} , B_{4} = -{{1}\over{30}} , B_{6} = {{1}\over{42}} , ...$
They occur in many diverse areas of mathematics including the series expansions of $\tan(x)$, Fermat's Last theorem, ...