Concrete mathematics : a foundation for computer science

322 GENERATING FUNCTIONS
middle of Table 321, and it’s also the special case m = 1 of (1, (","), (mzL),
(“,‘“), ), which appears further down; it’s also the special case c = 2 of
the closely related sequence (1, c, (‘:‘) I (‘12), . ). We can derive it from the
generating function for (1 , 1 , 1 , 1, . . ) by taking cumulative sums as in (7.21);
that is, by dividing 1 /(l-z) by (1 -z). Or we can derive it from (1 , 1 , 1 , 1, . ) OK, OK, I’m conby
differentiation, using vinced already
(7.17).
The sequence (1 , 0, 1 , 0, . ) is another one whose generating function can
be obtained in many ways. We can obviously derive the formula 1, zZn =
l/( 1 - z2) by substituting z2 for z in the identity t, Z” = l/( 1 - z); we can
also apply cumulative summation to the sequence (1, -1 , 1, -1, . . . ), whose
generating function is l/(1 $ z), getting l/(1 +z)(l - z) = l/(1 -2’). And
there’s also a third way, which is based on a general method for extracting
the even-numbered terms (gc , 0, g2, 0, g4,0, . . . ) of any given sequence: If we
add G(-z) to G(+z) we get
therefore
G(Z)+ G(-z) = t gn(l +(-1)")~" = 2x g,[n evenlz”;
n n
G(z) + G(-z)
2
= t g2n zLn .
n
The odd-numbered terms can be extracted in a similar way,
G(z) - G(-z)
2
=t g2n+1zZn+'
n
(7.22)
In the special case where g,, =I 1 and G(z) = l/( 1 -z), the generating function
for(1,0,1,0,...)is~(~(z)+~(-z))=t(&+&)=A.
Let’s try this extraction trick on the generating function for Fibonacci
numbers. We know that I., F,zn = z/( 1 - z - 2'); hence
t F2nz 2n = ;(j57+l+r’,)
n
1 ( 2 + 22 - 23 - 2 + z2 + z3 ) z2
=-
2 (I -z2)2-22 = l-322+24
This generates the sequence (Fo, 0, F2,0, F4,. . . ); hence the sequence of alternate
F’s, (Fo,Fl,Fd,F6,...) = (0,1,3,8,... ), has a simple generating function:
F2,,zn =
IL
n
z
l-3z+z2
(7.24)