Bern {DPQ}

Bernoulli Numbers

Description

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

Usage

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

Arguments

n

integer, n \ge 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
}