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Combinatorial Function
====================================================================
::

  f := operator 'f
 

   (1)  f
                                                          Type: BasicOperator

::
	
  D(product(f(i,x),i=1..m),x)
 

          m           m    f  (i,x)
        ++-++        --+    ,2
   (2)   | |   f(i,x)>     --------
         | |         --+    f(i,x)
        i= 1         i= 1
                                                     Type: Expression Integer


The binomial``(n, r)`` returns the number of subsets of r objects
taken among n objects, i.e. ``n!/(r! * (n-r)!)`` 

The binomial coefficients are the coefficients of the series expansion
of a power of a binomial, that is 

..    n
..   --+    / n \   k             n
..   >     |     | x    =  (1 + x)
..   --+    \ k /
..   k= 0

.. math::
  \sum_{k=0}^n \binom{n}{k} x^k = (1+x)^n

   
This leads to the famous pascal triangle. First we expose the OutputForm
domain, which is normally hidden, so we can use it to format the lines. ::

  )set expose add constructor OutputForm

Next we define a function that will output the list of binomial coefficients
right justified with proper spacing: ::

  pascalRow(n) == [right(binomial(n,i),4) for i in 0..n]

and now we format the whole line so that it looks centered: ::

  displayRow(n)==output center blankSeparate pascalRow(n) 

and we compute the triangle ::

  for i in 0..7 repeat displayRow i

giving the pretty result: ::
 
   Compiling function pascalRow with type NonNegativeInteger -> List 
      OutputForm 
   Compiling function displayRow with type NonNegativeInteger -> Void 
                                     1
                                  1    1
                                1    2    1
                             1    3    3    1
                           1    4    6    4    1
                        1    5   10   10    5    1
                      1    6   15   20   15    6    1
                   1    7   21   35   35   21    7    1

See Also:

* ``)show CombinatorialFunction``
* ``)d op binomial``
* ``)show OutputForm``
* ``)help set``