Concrete mathematics : a foundation for computer science

198 BINOMIAL COEFFICIENTS
Generating functions give us powerful ways to discover and/or prove
identities. For example, the binomial theorem tells us that (1 + z)~ is the
generating function for the sequence ((i) , (;) , (;) , . . ):
Similarly,
(1 +z)' = x (;)2
k30
(1 +z)” = x (;)zk.
k>O
If we multiply these togethe:r, we get another generating function:
(1 +z)T(l +z)S = (1 +z)'+s.
And now comes the punch line: Equating coefficients of z” on both sides of
this equation gives us
g:)(A) = (T).
We’ve discovered Vandermonde’s convolution, (5.27)! [5.27)! =
That was nice and easy; let’s try another. This time we use (1 -z)~, which
is the generating function for the sequence ((-1 )"(G)) = ((h) , -(;), (i) , . . . ).
Multiplying by (1 + z)~ gives another generating function whose coefficients
we know:
(1 -- z)'(l + z)' = (1 - z2)'.
Equating coefficients of z” now gives the equation
~(~)(n~k)t-lik = (-1)n12(~,)Inevenl. (5.55)
We should check this on a small case or two. When n = 3, for example,
the result is
(a)(;)-(F)(;)+(I)(T)-(;)(6) = O.
Each positive term is cancelled by a corresponding negative term. And the
same thing happens whenever n is odd, in which case the sum isn’t very
(5.27)[4.27)
(3.27)[2.27)
(1.27)(0.27)!.