sidef/scripts/bernoulli_numbers_recursive.sf

#!/usr/bin/ruby

# Recursive computation of Bernoulli numbers.

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

func bernoulli_number{};    # must be declared before used

func binomial(n, k) is cached {
    (k == 0) || (n == k) ? 1 : (binomial(n - 1, k - 1) + binomial(n - 1, k));
}

func bern_helper(n, k) {
    binomial(n, k) * (bernoulli_number(k) / (n - k + 1));
}

func bern_diff(n, k, d) {
    n < k ? d : bern_diff(n, k + 1, d - bern_helper(n + 1, k));
}

bernoulli_number = func(n) is cached {

    n.is_one && return 1/2;
    n.is_odd && return   0;

    n > 0 ? bern_diff(n - 1, 0, 1) : 1;
}

for i in range(0, 20, 2) {
    var num = bernoulli_number(i);
    printf("B(%2d) = %20s / %s\n", i, num.nude);
}