sidef-scripts/Math/euler_numbers_from_factorials.sf

#!/usr/bin/ruby

# Author: Daniel "Trizen" Șuteu
# Date: 22 May 2018
# https://github.com/trizen

# A new algorithm for computing the Euler numbers.

# See also:
#   https://oeis.org/A122045
#   https://en.wikipedia.org/wiki/Euler_number

func euler_numbers(n) {

    var E = [1]

    for i in (1 .. n) {
        for k in (0 ..^ i) {
            E[i] := 0 -= (E[k] / (2*(i-k))!)
        }
    }

    E.map_kv { |k,v| v * (2*k)! }
}

var E = euler_numbers(20)       # first 20 Euler numbers at even indices

for k,v in (E.kv) {
    say "E(#{2*k}) = #{v}"
}

__END__
E(0) = 1
E(2) = -1
E(4) = 5
E(6) = -61
E(8) = 1385
E(10) = -50521
E(12) = 2702765
E(14) = -199360981
E(16) = 19391512145
E(18) = -2404879675441
E(20) = 370371188237525
E(22) = -69348874393137901
E(24) = 15514534163557086905
E(26) = -4087072509293123892361
E(28) = 1252259641403629865468285
E(30) = -441543893249023104553682821
E(32) = 177519391579539289436664789665
E(34) = -80723299235887898062168247453281
E(36) = 41222060339517702122347079671259045
E(38) = -23489580527043108252017828576198947741
E(40) = 14851150718114980017877156781405826684425