Part 1:

use strict;
use warnings;

use List::Util qw( sum );

sub calcIt {
   our $r;
   local $r = sub {
      my ($depth) = @_;
      if ($depth == 0) {
         return sum 1..$_;
      } else {
         return sum map { $r->($depth-1) } 1..$_;
      }
   };

   return $r->(5) for $_[0];
}

print(calcIt(10), "\n");  # 11440
print(calcIt(20), "\n");  # 657800
[download]

Part 1, with memoization

use strict;
use warnings;

use List::Util qw( sum );

{
   my %memoize;

   sub calcIt {
      our $r;
      local $r = sub {
         my ($depth) = @_;
         return $memoize{"$depth:$_"} ||= (
            ($depth
               ? sum map { $r->($depth-1) } 1..$_
               : $_ * ( $_ + 1 ) / 2
            )
         );
      };
      return $r->(5) for $_[0];
   }
}

print(calcIt(10), "\n");  # 11440
print(calcIt(20), "\n");  # 657800
[download]

Note: I changed 0.. to 1... You were adding way too many zeros.