The hackery with the short-circuit sub calls is part of the problem. If I rewrite this in a "naive" way:

sub A
{
    my ($m, $n) = @_;
    return 0    if $n == 0; 
    return 2*$n if $n >  0 and $m == 0;
    return 2    if $n == 1 and $m >  0;
    A($m-1, A($m, $n-1));
}

print A ($ARGV[0], $ARGV[1]), "\n";
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I get a reasonable answer with (3,3). However, on (4,4), I get "Not a CODE reference at a.pl line 7" followed by a memory fault. (I'm using 5.6.1 on HP-UX, BTW.)