use strict;
use warnings;

my ($n, $k) = (20, 11);

print "S($n, $k) = ", S_formulaic($n, $k), "\n";
print "S($n, $k) = ", S_recursive($n, $k), "\n";

sub S_recursive
{
    my ($n, $k) = @_;

    return 1 if $n == 0 && $k == 0;
    return 0 if $k == 0;
    return 1 if $k == 1;
    return 1 if $n == $k;

    return S_recursive($n - 1, $k - 1) + $k * S_recursive($n - 1, $k);
}

sub S_formulaic
{
    my ($n, $k) = @_;
    my  $sum    = 0;

    for my $j (0 .. $k)
    {
        $sum += (-1) ** ($k - $j) * C($k, $j) * $j ** $n;
    }

    return (1 / fact($k)) * $sum;
}

sub C
{
    my ($n, $k) = @_;

    my $p  = 1;
       $p *= $_ for ($n - $k + 1) .. $n;

    return $p / fact($k);
}

sub fact
{
    my ($n) = @_;
    my  $p  = 1;
        $p *= $_ for 2 .. $n;

    return $p;
}
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Output:

17:17 >perl 1740_SoPW.pl
S(20, 11) = 1900842429486
S(20, 11) = 1900842429486

17:17 >
[download]