If you add a column of 1's on the left (and a column of sums of 1 .. N after the 1..N column), you get Pascal's triangle, and thus the link to combinatorics: 

1   1   1    1     1     1     1     1     1     1
1   2   3    4     5     6     7     8     9     10
1   3   6    10    15    21    28    36    45    55
1   4   10   20    35    56    84    120   165   220
1   5   15   35    70    126   210   330   495   715
1   6   21   56    126   252   462   792   1287  2002
1   7   28   84    210   462   924   1716  3003  5005
1   8   36   120   330   792   1716  3432  6435  11440

[download]

Update: added missing 3rd column as per eric256's observation

Fcn(D,N) => (5,10) seems to correspond to Choose(16,7) where Choose(n,k) is n!/k!(n-k)!

Likewise, Fcn(5,20) seems to correspond to Choose(26,7).

More generically, it looks like your function can be reduced to Choose(D+N+1,D+2)

Of course, you still have recursion in the computation of the factorial, so I don't think you can quite avoid it entirely.

-xdg

Code written by xdg and posted on PerlMonks is public domain. It is provided as is with no warranties, express or implied, of any kind. Posted code may not have been tested. Use of posted code is at your own risk.

In reply to Re^2: Math fun. by xdg
in thread Math fun. by BrowserUk

Title:
Use:  <p> text here (a paragraph) </p>
and:  <code> code here </code>
to format your post, it's "PerlMonks-approved HTML":


  • Posts are HTML formatted. Put <p> </p> tags around your paragraphs. Put <code> </code> tags around your code and data!
  • Titles consisting of a single word are discouraged, and in most cases are disallowed outright.
  • Read Where should I post X? if you're not absolutely sure you're posting in the right place.
  • Please read these before you post! —
  • Posts may use any of the Perl Monks Approved HTML tags:
    a, abbr, b, big, blockquote, br, caption, center, col, colgroup, dd, del, details, div, dl, dt, em, font, h1, h2, h3, h4, h5, h6, hr, i, ins, li, ol, p, pre, readmore, small, span, spoiler, strike, strong, sub, summary, sup, table, tbody, td, tfoot, th, thead, tr, tt, u, ul, wbr
  • You may need to use entities for some characters, as follows. (Exception: Within code tags, you can put the characters literally.)
            For:     Use:
    & &amp;
    < &lt;
    > &gt;
    [ &#91;
    ] &#93;
  • Link using PerlMonks shortcuts! What shortcuts can I use for linking?
  • See Writeup Formatting Tips and other pages linked from there for more info.