repellent:

Sweet! So of course I had to take another look:

sub pascal_tri_row {
    my $r = shift;
    return () if $r < 0;

    my @row = (1) x ($r + 1);

    for my $i (1 .. $r - 1)
    {
        $row[$_] += $row[$_ - 1]
          for reverse 1 .. $i;
    }
    return @row;
}

sub robo_2 {
    my $row = shift;
    my @cols = (1);
    ++$row;
    $cols[$_] = $cols[$_-1] * ($row-$_)/$_ for 1 .. $row-1;
    return @cols;
}

sub triangle {
    my $numTosses = shift;

    my @triangle = (0, 1, 0);
    for (1 .. $numTosses) {
        my @newTriangle=(0);
        push @newTriangle, $triangle[$_]+$triangle[$_+1] for 0 .. $#tr
+iangle-1;
        push @newTriangle, 0;
        @triangle = @newTriangle;
    }

    return @triangle[1..$#triangle-1];
}

use Benchmark qw(cmpthese);

print "robo_1: ", join(" ",triangle(8)), "\n";
print "repel1: ", join(" ",pascal_tri_row(8)), "\n";
print "robo_2: ", join(" ",robo_2(8)), "\n";

cmpthese -1, {
    robo_tri  => sub { triangle(32) },
    repel_tri => sub { pascal_tri_row(32) },
    robo_2    => sub { robo_2(32) },
};
[download]

First I put in a trace so I could be sure I wasn't generating useless values. Next, I couldn't bear the zeroes you had to add to your function to match my original return value. They were just sentinel values to simplify the calculation, anyway. So I fixed the original to return only the values of interest. Finally, I had to squeeze a little more speed out of it:

$ perl 892898.pl
robo_1: 1 8 28 56 70 56 28 8 1
repel1: 1 8 28 56 70 56 28 8 1
robo_2: 1 8 28 56 70 56 28 8 1
             Rate  robo_tri repel_tri    robo_2
robo_tri   2196/s        --      -42%      -93%
repel_tri  3775/s       72%        --      -88%
robo_2    31946/s     1355%      746%        --
[download]

You should've read a bit further down in the wikipedia article you linked. It had a much better algorithm for calculating the coefficients of any row. It saved a nested loop.

;^)

...roboticus

When your only tool is a hammer, all problems look like your thumb.