bernoulli_numbers_recurrence.sf

#!/usr/bin/ruby

# A recurrence for computing Bernoulli numbers.

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

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

# See also:
#   https://en.wikipedia.org/wiki/Bernoulli_number#Recursive_definition

func a((0)) { 1 }

func a(n) is cached {
    1 - sum(0..^n, {|k| binomial(n, k) * a(k) / (n - k + 1) })
}

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