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bernoulli_numbers_from_factorials_visual.pl
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bernoulli_numbers_from_factorials_visual.pl
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#!/usr/bin/perl
# Author: Daniel "Trizen" Șuteu
# Date: 02 December 2017
# https://github.com/trizen
# A new algorithm for computing Bernoulli numbers (visualization).
# Inspired from Norman J. Wildberger video lecture:
# https://www.youtube.com/watch?v=qmMs6tf8qZ8
# See also:
# https://en.wikipedia.org/wiki/Bernoulli_number#Connection_with_Pascal’s_triangle
use 5.010;
use strict;
use warnings;
use Math::AnyNum qw(:overload factorial bernfrac);
sub bernoulli_numbers {
my ($n) = @_;
my @B = (1, (0) x $n);
foreach my $i (1 .. $n) {
if ($i % 2 != 0 and $i > 1) {
## next;
}
foreach my $k (0 .. $i - 1) {
if ($k % 2 != 0 and $k > 1) {
## next;
}
my $f = factorial($i - $k + 1);
my $d = $B[$i] - $B[$k] / $f;
printf("[%2s, %s] -> %6s / %2s! - %6s / %s! / %2s! = %6s / %2s!\n",
$i, $k, $B[$i] * factorial($i),
$i, $B[$k] * factorial($k),
$k,
$i - $k + 1,
$d * factorial($i), $i);
$B[$i] = $d;
}
say '';
}
map { $B[$_] * factorial($_) } 0 .. $#B;
}
my @B = bernoulli_numbers(10); # first 10 Bernoulli numbers
foreach my $i (0 .. $#B) {
# Verify the results
if ($i > 1 and $B[$i] != bernfrac($i)) {
die "error for i=$i";
}
say "B($i) = $B[$i]";
}
__END__
[ 1, 0] -> 0 / 1! - 1 / 0! / 2! = -1/2 / 1!
[ 2, 0] -> 0 / 2! - 1 / 0! / 3! = -1/3 / 2!
[ 2, 1] -> -1/3 / 2! - -1/2 / 1! / 2! = 1/6 / 2!
[ 3, 0] -> 0 / 3! - 1 / 0! / 4! = -1/4 / 3!
[ 3, 1] -> -1/4 / 3! - -1/2 / 1! / 3! = 1/4 / 3!
[ 3, 2] -> 1/4 / 3! - 1/6 / 2! / 2! = 0 / 3!
[ 4, 0] -> 0 / 4! - 1 / 0! / 5! = -1/5 / 4!
[ 4, 1] -> -1/5 / 4! - -1/2 / 1! / 4! = 3/10 / 4!
[ 4, 2] -> 3/10 / 4! - 1/6 / 2! / 3! = -1/30 / 4!
[ 4, 3] -> -1/30 / 4! - 0 / 3! / 2! = -1/30 / 4!
[ 5, 0] -> 0 / 5! - 1 / 0! / 6! = -1/6 / 5!
[ 5, 1] -> -1/6 / 5! - -1/2 / 1! / 5! = 1/3 / 5!
[ 5, 2] -> 1/3 / 5! - 1/6 / 2! / 4! = -1/12 / 5!
[ 5, 3] -> -1/12 / 5! - 0 / 3! / 3! = -1/12 / 5!
[ 5, 4] -> -1/12 / 5! - -1/30 / 4! / 2! = 0 / 5!
[ 6, 0] -> 0 / 6! - 1 / 0! / 7! = -1/7 / 6!
[ 6, 1] -> -1/7 / 6! - -1/2 / 1! / 6! = 5/14 / 6!
[ 6, 2] -> 5/14 / 6! - 1/6 / 2! / 5! = -1/7 / 6!
[ 6, 3] -> -1/7 / 6! - 0 / 3! / 4! = -1/7 / 6!
[ 6, 4] -> -1/7 / 6! - -1/30 / 4! / 3! = 1/42 / 6!
[ 6, 5] -> 1/42 / 6! - 0 / 5! / 2! = 1/42 / 6!
[ 7, 0] -> 0 / 7! - 1 / 0! / 8! = -1/8 / 7!
[ 7, 1] -> -1/8 / 7! - -1/2 / 1! / 7! = 3/8 / 7!
[ 7, 2] -> 3/8 / 7! - 1/6 / 2! / 6! = -5/24 / 7!
[ 7, 3] -> -5/24 / 7! - 0 / 3! / 5! = -5/24 / 7!
[ 7, 4] -> -5/24 / 7! - -1/30 / 4! / 4! = 1/12 / 7!
[ 7, 5] -> 1/12 / 7! - 0 / 5! / 3! = 1/12 / 7!
[ 7, 6] -> 1/12 / 7! - 1/42 / 6! / 2! = 0 / 7!
[ 8, 0] -> 0 / 8! - 1 / 0! / 9! = -1/9 / 8!
[ 8, 1] -> -1/9 / 8! - -1/2 / 1! / 8! = 7/18 / 8!
[ 8, 2] -> 7/18 / 8! - 1/6 / 2! / 7! = -5/18 / 8!
[ 8, 3] -> -5/18 / 8! - 0 / 3! / 6! = -5/18 / 8!
[ 8, 4] -> -5/18 / 8! - -1/30 / 4! / 5! = 17/90 / 8!
[ 8, 5] -> 17/90 / 8! - 0 / 5! / 4! = 17/90 / 8!
[ 8, 6] -> 17/90 / 8! - 1/42 / 6! / 3! = -1/30 / 8!
[ 8, 7] -> -1/30 / 8! - 0 / 7! / 2! = -1/30 / 8!
[ 9, 0] -> 0 / 9! - 1 / 0! / 10! = -1/10 / 9!
[ 9, 1] -> -1/10 / 9! - -1/2 / 1! / 9! = 2/5 / 9!
[ 9, 2] -> 2/5 / 9! - 1/6 / 2! / 8! = -7/20 / 9!
[ 9, 3] -> -7/20 / 9! - 0 / 3! / 7! = -7/20 / 9!
[ 9, 4] -> -7/20 / 9! - -1/30 / 4! / 6! = 7/20 / 9!
[ 9, 5] -> 7/20 / 9! - 0 / 5! / 5! = 7/20 / 9!
[ 9, 6] -> 7/20 / 9! - 1/42 / 6! / 4! = -3/20 / 9!
[ 9, 7] -> -3/20 / 9! - 0 / 7! / 3! = -3/20 / 9!
[ 9, 8] -> -3/20 / 9! - -1/30 / 8! / 2! = 0 / 9!
[10, 0] -> 0 / 10! - 1 / 0! / 11! = -1/11 / 10!
[10, 1] -> -1/11 / 10! - -1/2 / 1! / 10! = 9/22 / 10!
[10, 2] -> 9/22 / 10! - 1/6 / 2! / 9! = -14/33 / 10!
[10, 3] -> -14/33 / 10! - 0 / 3! / 8! = -14/33 / 10!
[10, 4] -> -14/33 / 10! - -1/30 / 4! / 7! = 19/33 / 10!
[10, 5] -> 19/33 / 10! - 0 / 5! / 6! = 19/33 / 10!
[10, 6] -> 19/33 / 10! - 1/42 / 6! / 5! = -14/33 / 10!
[10, 7] -> -14/33 / 10! - 0 / 7! / 4! = -14/33 / 10!
[10, 8] -> -14/33 / 10! - -1/30 / 8! / 3! = 5/66 / 10!
[10, 9] -> 5/66 / 10! - 0 / 9! / 2! = 5/66 / 10!
B(0) = 1
B(1) = -1/2
B(2) = 1/6
B(3) = 0
B(4) = -1/30
B(5) = 0
B(6) = 1/42
B(7) = 0
B(8) = -1/30
B(9) = 0
B(10) = 5/66