# Category:Bernoulli Numbers

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This category contains results about Bernoulli Numbers.

Definitions specific to this category can be found in Definitions/Bernoulli Numbers.

The **Bernoulli numbers** $B_n$ are a sequence of rational numbers defined by:

### Generating Function

- $\displaystyle \frac x {e^x - 1} = \sum_{n \mathop = 0}^\infty \frac {B_n x^n} {n!}$

### Recurrence Relation

- $B_n = \begin{cases} 1 & : n = 0 \\ \displaystyle - \sum_{k \mathop = 0}^{n - 1} \binom n k \frac {B_k} {n + 1 - k} & : n > 0 \end{cases}$

or equivalently:

- $B_n = \begin{cases} 1 & : n = 0 \\ \displaystyle - \frac 1 {n+1} \sum_{k \mathop = 0}^{n - 1} \binom {n+1} k B_k & : n > 0 \end{cases}$

## Subcategories

This category has the following 4 subcategories, out of 4 total.

## Pages in category "Bernoulli Numbers"

The following 18 pages are in this category, out of 18 total.

### P

- Power Series Expansion for Cosecant Function
- Power Series Expansion for Cotangent Function
- Power Series Expansion for Hyperbolic Cosecant Function
- Power Series Expansion for Hyperbolic Cotangent Function
- Power Series Expansion for Hyperbolic Tangent Function
- Power Series Expansion for Tangent Function

### S

- Sum of Bernoulli Numbers by Binomial Coefficients Vanishes
- Sum of Bernoulli Numbers by Power of Two and Binomial Coefficient
- Sum of Reciprocals of Even Powers of Odd Integers
- Sum of Reciprocals of Even Powers of Odd Integers/Corollary
- Sum of Reciprocals of Powers of Integers Alternating in Sign
- Sum of Reciprocals of Powers of Integers Alternating in Sign/Corollary