An Upside-Down Pascal's Triangle

Take Pascal's triangle, but instead of putting ( nk ) in each entry, put the reciprocal of  (n + 1) ( nk ).

Then each number is the sum of the two numbers below it:

                1
           1/2     1/2
       1/3     1/6     1/3
   1/4    1/12    1/12    1/4
1/5   1/20    1/30    1/20   1/5
        ... and so on.

I think it's a bit surprising that there's an addition formula for reciprocals at all.