sidef-scripts/Math/bernoulli_numbers_recurrence_2.sf

#!/usr/bin/ruby

# Daniel "Trizen" Șuteu
# Date: 24 February 2018
# https://github.com/trizen

# A new recurrence for computing Bernoulli numbers.

# Formula:
#  a(0) = 1
#  a(n) = 1 - ((Sum_{k=0..n-1} a(k) * binomial(n + 1, k)) / (n+1))

# Which gives us the nth-Bernoulli number, B_n, as:
#   B_n = a(n)

# See also:
#   https://en.wikipedia.org/wiki/Bernoulli_number
#   https://arxiv.org/pdf/0708.0809.pdf (Compositional Bernoulli numbers)

func a((0)) { 1 }

func a(n) is cached {
    1 - (sum(^n, {|k| a(k) * binomial(n+1, k) }) / (n+1))
}

for n in (0..60 `by` 2) {
    printf("B(%2d) = %50s / %s\n", n, a(n) -> nude)
}