In mathematics, the Bernoulli numbers Bn were first discovered in connection with the closed forms of the sums
Bernoulli numbers may be calculated by using the following recursive formula:
The Bernoulli numbers may also be defined using the technique of generating functions. Their exponential generating function is x/(ex - 1), so that:
Sometimes the lower-case bn is used in order to distinguish these from the Bell numbers.
The first few Bernoulli numbers are listed below.
It can be shown that Bn = 0 for all odd n other than 1. The appearance of the peculiar value B12 = -691/2730 appears to rule out the possibility of a simple closed form for Bernoulli numbers.
The Bernoulli numbers also appear in the Taylor series expansion of the tangent and hyperbolic tangent functions, in the Euler-Maclaurin formula, and in expressions of certain values of the Riemann zeta function.
In note G of Ada Byron's notes on the analytical engine from 1842 an algorithm for computer generated Bernoulli numbers was described for the first time.