Concrete mathematics : a foundation for computer science

336 GENERATING FUNCTIONS
This is a recurrence that “goes all the way back” from f,-l through all previous
values, so it’s different from the other recurrences we’ve seen so far
in this chapter. We used a special method to get rid of a similar right-side
sum in Chapter 2, when we solved the quicksort recurrence (2.12); namely,
we subtracted one instance of the recurrence from another (f,+l - fn). This
trick would get rid of the t now, as it did then; but we’ll see that generating
functions allow us to work directly with such sums. (And it’s a good thing
that they do, because we will be seeing much more complicated recurrences
before long.)
Step 1 is finished; Step :2 is where we need to do a new thing:
F(z) = tf,zn = tf,,zn+tfkzn[kO]zn
n n kn n
= zF(z) + ~fkZk~[n>k]Znpk + ez
k n
= zF(z) + F(z) 1 zm + &
m>O
= zF(z) + F(z) & + it-.
1-z
The key trick here was to change zn to zk zn-k; this made it possible to express
the value of the double sum in terms of F(z), as required in Step 2.
Now Step 3 is simple algebra, and we find
F(z) = ’
1 -3zf22 *
Those of us with a zest for memorization will recognize this as the generating
function (7.24) for the even-numbered Fibonacci numbers. So, we needn’t go
through Step 4; we have found a somewhat surprising answer to the spansof-fans
problem:
fn = F2n 1 for n 3 0. ( 7.42)
7.4 SPECIAL GENERATING FUNCTIONS
Step 4 of the four-step procedure becomes much easier if we know
the coefficients of lots of diff’erent power series. The expansions in Table 321
are quite useful, as far as they go, but many other types of closed forms are
possible. Therefore we ought to supplement that table with another one,
which lists power series that correspond to the “special numbers” considered
in Chapter 6.