Mathematics for Computer Science

“mcs” — 2017/3/3 — 11:21 — page 700 — #708
700
Chapter 16
Generating Functions
Let T n be the number of different combinations of n mishaps Dan can suffer in one
day. For example, T 3 D 7, because there are seven possible combinations of three
mishaps:
spills 0 1 0 1 1 0 0
falls 0 0 1 1 0 1 0
stubs 3 2 2 1 0 0 1
blurts 0 0 0 0 2 2 2
(a) Express the generating function
as a quotient of polynomials.
T .x/ WWD T 0 C T 1 x C T 2 x 2 C
(b) Put integers in the boxes that make this equation true:
g.x/ D 1 x
C .1 x/
2
(c) Write a closed-form expression for T n :
Problems for Section 16.4
Practice Problems
Problem 16.13.
Let b, c, a 0 , a 1 , a 2 ,. . . be real numbers such that
a n D b.a n
1 / C c
for n 1.
Let G.x/ be the generating function for this sequence.
(a) Express the coefficient of x n for n 1 in the series expansion of bxG.x/ in
terms of b and a i for suitable i.