R: Bernoulli Numbers

Bern {DPQ} | R Documentation

## Bernoulli Numbers

### Description

Return the n-th Bernoulli number B_n, (or Bn+, see the reference), where B_1 = + 1/2.

### Usage

    Bern(n, verbose = getOption("verbose", FALSE))

### Arguments

`n` | integer, n >= 0.
`verbose` | logical indicating if computation should be traced.

### Value

The number B_n of type `numeric`.

A side effect is the caching of computed Bernoulli numbers in the hidden `environment` `.bernoulliEnv`.

### Author(s)

Martin Maechler

### References

https://en.wikipedia.org/wiki/Bernoulli_number

### See Also

`Bernoulli` in Rmpfr in arbitrary precision via Riemann's zeta function.

The next version of package gmp is to contain `BernoulliQ()`, providing exact Bernoulli numbers as big rationals (class `"bigq"`).

### Examples

    (B.0.10 <- vapply(0:10, Bern, 1/2))
    ## [1]  1.00000000 +0.50000000  0.16666667  0.00000000 -0.03333333  0.00000000
    ## [7]  0.02380952  0.00000000 -0.03333333  0.00000000  0.07575758
    if(requireNamespace("MASS")) {
      print( MASS::fractions(B.0.10) )
      ## 1  +1/2   1/6    0  -1/30     0  1/42     0 -1/30     0  5/66
    }

[Package DPQ version 0.5-2 Index]