Mathematics for Computer Science

“mcs” — 2017/3/3 — 11:21 — page 680 — #688
680
Chapter 16
Generating Functions
16.2.4 Counting Donuts with the Convolution Rule
We can use the Convolution Rule to derive in another way the generating function
D.x/ for the number of ways to select chocolate and plain donuts given in (16.6).
To begin, there is only one way to select exactly n chocolate donuts. That means
every coefficient of the generating function for selecting n chocolate donuts equals
one. So the generating function for chocolate donut selections is 1=.1 x/; likewise
for the generating function for selecting only plain donuts. Now by the Convolution
Rule, the generating function for the number of ways to select n donuts when both
chocolate and plain flavors are available is
D.x/ D 1
1 x 1
1 x D 1
.1 x/ 2 :
So we have derived (16.6) without appeal to (16.5).
Our application of the Convolution Rule for two flavors carries right over to the
general case of k flavors; the generating function for selections of donuts when k
flavors are available is 1=.1 x/ k . We already derived the formula for the number
nC.k 1/
of ways to select a n donuts when k flavors are available, namely,
n from
Corollary 15.5.3. So we have
!
Œx n 1 n C .k 1/
D : (16.10)
.1 x/ k n
Extracting Coefficients from Maclaurin’s Theorem
We’ve used a donut-counting argument to derive the coefficients of 1=.1 x/ k ,
but it’s instructive to derive this coefficient algebraically, which we can do using
Maclaurin’s Theorem:
Theorem 16.2.1 (Maclaurin’s Theorem).
f .x/ D f .0/ C f 0 .0/x C f 00 .0/
2Š
x 2 C f 000 .0/
3Š
x 3 C C f .n/ .0/
x n C :
nŠ
This theorem says that the nth coefficient of 1=.1 x/ k is equal to its nth derivative
evaluated at 0 and divided by nŠ. Computing the nth derivative turns out not to
be very difficult
d n
d n x
1
.kCn/
D k.k C 1/ .k C n 1/.1 x/
.1 x/ k