Concrete mathematics : a foundation for computer science

324 GENERATING FUNCTIONS
Fortunately the problem is easy to fix, since we can add [n = 11 to the right;
this adds 1 when n = 1, and it makes no change when n # 1. So, we have
gn = s-1 +a-2+[n=ll;
this is the equation called fo:r in Step 1.
Step 2 now asks us to t:ransform the equation for (g,,) into an equation
for G(z) = t, gnzn. The task is not difficult:
G(z) = x gnzn = ~gnlzn+tg,~rzn+~[n=l]zn
n
= ;gnzn+l+;gnzn+2 fnz
n n
= G(z) + z’G(z) + z.
Step 3 is also simple in this case; we have
G(z) = '
l-z-z2'
which of course comes as no surprise.
Step 4 is the clincher. We carried it out in Chapter 6 by having a sudden
flash of inspiration; let’s go more slowly now, so that we can get through
Step 4 safely later, when we meet problems that are more difficult. What is
b”l z
l-z-22'
the coefficient of zn when z/( 1 - z - z2) is expanded in a power series? More
generally, if we are given any rational function
P(z)
R(z) = Qo,
where P and Q are polynomials, what is the coefficient [z”] R(z)?
There’s one kind of rational function whose coefficients are particularly
nice, namely
(1 - puz)m+1
= x (m;n)ap"z"
n30
(7.25)
(The case p = 1 appears in Table 321, and we can get the general formula
shown here by substituting pz for z.) A finite sum of functions like (7.25),
s(z) = (1 - pyl,-,+, '-
a2
(1 -p2Z)m2+' +'.'+ (1 -pLZ)mL+l '
al
(7.4