In mathematics, the **Bernoulli numbers** *B*_{n} were first discovered in connection with the closed forms of the sums

The Bernoulli numbers were first studied by Jakob Bernoulli, after whom they were named by Abraham de Moivre.

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*/(*e ^{x}* - 1), so that:

Sometimes the lower-case *b _{n}* is used in order to distinguish these from the Bell numbers.

The first few Bernoulli numbers are listed below.

n | B_{n} |
---|---|

0 | 1 |

1 | -1/2 |

2 | 1/6 |

3 | 0 |

4 | -1/30 |

5 | 0 |

6 | 1/42 |

7 | 0 |

8 | -1/30 |

9 | 0 |

10 | 5/66 |

11 | 0 |

12 | -691/2730 |

13 | 0 |

14 | 7/6 |

It can be shown that *B*_{n} = 0 for all odd *n* other than 1.
The appearance of the peculiar value *B _{12}* = -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.