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

*n*. The closed forms are always polynomials in

*m*of degree

*n*+1 and are called

**Bernoulli polynomials**. The coefficients of the Bernoulli polynomials are closely related to the Bernoulli numbers, as follows:

*n*to be 1, we have 0 + 1 + 2 + ... + (

*m*-1) = 1/2 (

*B*

_{0}

*m*

^{2}+ 2

*B*

_{1}

*m*

^{1}) = 1/2 (

*m*

^{2}-

*m*).

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:

*B*

_{0}= 1.

The Bernoulli numbers may also be defined using the technique of generating functions.
Their exponential generating function is *x*/(*e ^{x}* - 1), so that:

*x*of absolute value less than 2π (2π is the radius of convergence of this power series).

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.