Is this of any help?

$n\$d   1       2       3       4       5       6       7
1       1       1       1       1       1       1       1
2       4       5       6       7       8       9       10
3       10      15      21      28      36      45      55
4       20      35      56      84      120     165     220
5       35      70      126     210     330     495     715
6       56      126     252     462     792     1287    2002
7       84      210     462     924     1716    3003    5005
8       120     330     792     1716    3432    6435    11440
9       165     495     1287    3003    6435    12870   24310
10      220     715     2002    5005    11440   24310   48620
11      286     1001    3003    8008    19448   43758   92378
12      364     1365    4368    12376   31824   75582   167960
13      455     1820    6188    18564   50388   125970  293930
14      560     2380    8568    27132   77520   203490  497420
15      680     3060    11628   38760   116280  319770  817190
16      816     3876    15504   54264   170544  490314  1307504
17      969     4845    20349   74613   245157  735471  2042975
18      1140    5985    26334   100947  346104  1081575 3124550
19      1330    7315    33649   134596  480700  1562275 4686825
20      1540    8855    42504   177100  657800  2220075 6906900
[download]

The above was produced using the following program:

use strict;
use warnings;

use List::Util qw( sum );

{
   my %memoize;

   sub calcIt {
      my ($n, $d) = @_;
      return $memoize{"$n:$d"} ||= (
         ($d
            ? sum map { calcIt($_, $d-1) } 1..$n
            : $n * ( $n + 1 ) / 2
         )
      );
   }
}

{
   my $N = 20;
   my $D = 7;

   print('$n\\$d');
   for my $d (1..$D) {
      print("\t$d");
   }
   print("\n");

   for my $n (1..$N) {
      print("$n");
      for my $d (1..$D) {
         print("\t", calcIt($n, $d));
      }
      print("\n");
   }
}
[download]

You can really appreciate the effect of memoization here.