I suspect you need to start $toggle reversed for odd values

Indeed. Using my $toggle = $P & 1 ? 1 : -1;, the output makes much more sense:

you don't show most of the code so I didn't go much further in looking

The entire code looks like this:

#! perl -slw
use strict;

sub PTn {
    my @row;
    for( 1 .. shift ) {
        push @row, 1;
        $row [$_] += $row [$_ - 1] for reverse 1 .. @row - 2;
    }
    return @row;
}

sub expN {
    my( $P, $N, $X ) = @_;
    my $xDIVn = $X / $N;
    my @coefs = PTn( $P+1 );
    my $rv = 0;
    my $toggle = $P & 1 ? 1 : -1;
    for my $p ( reverse 1 .. $P ) {
        $rv += $toggle * $coefs[ $p ] * $N / $p * exp( -$p * $xDIVn );
        $toggle *= -1;
    }
    return $rv + $X;
}

our $H //= 10;
our $B //= 32;

print "Using N x 2**$B vectors:";

for my $inserts ( map{ 2**$_*3/4, 2**$_ } 4 .. $B ) {
    printf "%10d : ", $inserts;
    for my $h ( 1 .. $H ) {
        printf "%6d ", expN( $h, 2**$B, $inserts ) - expN( $h, 2**$B, 
+0 );
    }
    print '';
}
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