sidef-scripts/Math/harmonic_numbers_closed_form.sf

#!/usr/bin/ruby

# A remarkable closed-form to the n-th Harmonic number:
#   H_n = ((binomial(n + (n+1)!, n) - 1) / (n+1)!) - (n+1)*floor(((binomial(n + (n+1)!, n) - 1) / (n+1)!) / (n+1))

# Can it be computed efficiently?

# See also:
#   https://en.wikipedia.org/wiki/Harmonic_number
#   https://math.stackexchange.com/questions/52572/do-harmonic-numbers-have-a-closed-form-expression

# Formula posted by nczksv on math.stackexchange.com (5th April 2018):
#   https://math.stackexchange.com/a/2723193

func H(n) {
    with ((binomial(n + (n+1)!, n) - 1) / (n+1)!) {|g|
        g - (n+1)*floor(g / (n+1))
    }
}

say 10.of(H)    #=> [0, 1, 3/2, 11/6, 25/12, 137/60, 49/20, 363/140, 761/280, 7129/2520]