perl-scripts/Math/cauchy_numbers_of_first_type.pl

#!/usr/bin/perl

# Author: Daniel "Trizen" Șuteu
# Date: 09 February 2018
# https://github.com/trizen

# A new algorithm for computing the Cauchy numbers of first type.

# See also:
#   https://oeis.org/A006232    (numerators)
#   https://oeis.org/A006233    (denominators)

use 5.010;
use strict;
use warnings;

use Math::AnyNum qw(:overload factorial);

sub cauchy_numbers {
    my ($n) = @_;

    my @C = (1);

    foreach my $i (1 .. $n) {
        foreach my $k (0 .. $i - 1) {
            $C[$i] -= $C[$k] / ($i - $k + 1);
        }
    }

    map { (-1)**$_ * $C[$_] * factorial($_) } 0 .. $#C;
}

my @cauchy = cauchy_numbers(30);

foreach my $i (0 .. $#cauchy) {
    printf "C(%2d) = %40s / %s\n", $i, $cauchy[$i]->nude;
}

__END__
C( 0) =                                        1 / 1
C( 1) =                                        1 / 2
C( 2) =                                       -1 / 6
C( 3) =                                        1 / 4
C( 4) =                                      -19 / 30
C( 5) =                                        9 / 4
C( 6) =                                     -863 / 84
C( 7) =                                     1375 / 24
C( 8) =                                   -33953 / 90
C( 9) =                                    57281 / 20
C(10) =                                 -3250433 / 132
C(11) =                                  1891755 / 8
C(12) =                             -13695779093 / 5460
C(13) =                              24466579093 / 840
C(14) =                            -132282840127 / 360
C(15) =                             240208245823 / 48
C(16) =                         -111956703448001 / 1530
C(17) =                            4573423873125 / 4
C(18) =                       -30342376302478019 / 1596
C(19) =                        56310194579604163 / 168
C(20) =                    -12365722323469980029 / 1980
C(21) =                    161867055619224199787 / 1320
C(22) =                 -20953816286242674495191 / 8280
C(23) =                   4380881778942163832799 / 80
C(24) =             -101543126947618093900697699 / 81900
C(25) =              192060902780872132330221667 / 6552
C(26) =            -1092286933245454564213092649 / 1512
C(27) =             2075032177476967189228515625 / 112
C(28) =         -1718089509598695642524656240811 / 3480
C(29) =          1092041494691940355778302728249 / 80
C(30) =     -44810233755305010150728029810063187 / 114576