The easiest way to process data structures of variable depth and complexity is recursion:

#! perl -slw
use strict;

sub weightedSum {
    my( $r, $weight ) = ( @_, 1 );
    my $sum = 0;
    for ( @$r ) {
        if( ref ) {
            $sum += weightedSum( $_, $weight + 1 );
        }
        else {
            $sum += $_ * $weight;
        }
    }
    return $sum;
}

my $arrref = [
    [1,1,1,1],
    [1,1, [2,2] ],
    [1, [2, [3,3] ] ]
];

print weightedSum( $arrref );
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Produces:

C:\test>1152751
56
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A variation:

#! perl -slw
use strict;
use List::Util qw[ reduce ]; $a = $b; ## silence warning

sub weightedSum {
    my( $r, $weight ) = ( @_, 1 );
    return reduce{
        $a += ( ref( $b ) ) ? weightedSum( $b, $weight + 1 ): $b * $we
+ight
    } 0, @$r;
}

my $arrref = [
    [1,1,1,1],
    [1,1, [2,2] ],
    [1, [2, [3,3] ] ]
];

print weightedSum( $arrref );
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