Source code in the phix programming language

-- FreeBASIC said:
--'  1   1   1   1   1   1  
--'  1   2   3   4   5   6  
--'  1   3   6  10  15  21  
--'  1   4  10  20  35  56  
--'  1   5  15  35  70 126  
--'  1   6  21  56 126 252  
--' The Pascal triangle is rotated 45 deg.
--' to find the Catalan number we need to follow the diagonal
--' for top left to bottom right 
--' take the number on diagonal and subtract the number in de cell
--' one up and one to right 
--' 1 (2 - 1), 2 (6 - 4), 5 (20 - 15) ... 
--
-- The first thing that struck me was it is twice as big as it needs to be, 
--  something like this would do...
--    1   1   1   1   1   1  
--        2   3   4   5   6  
--            6  10  15  21  
--               20  35  56  
--                   70 126  
--                      252  
-- It is more obvious from the upper square that the diagonal on that, which is 
--  that same as column 1 on this, is twice the previous, which on the second 
--  diagram is in column 2. Further, once we have calculated the value for column 
--  one above, we can use it immediately to calculate the next catalan number and 
--  do not need to store it. Lastly we can overwrite row 1 with row 2 etc in situ, 
--  and the following shows what we need for subsequent rounds:
--    1   1   1   1   1
--    3   4   5   6  
--   10  15  21  
--   35  56  
--  126  (unused)