Concrete mathematics : a foundation for computer science

362 GENERATING FUNCTIONS
29 What is the sum of Fibonacci products
m>O k, +k>+...+k,=n
kl ,kz....,k,>O
30 If the generating function G(z) = l/( 1 - 1x2)(1 - (3~) has the partial
fraction decomposition a/( 1 -KZ) +b/( 1 - (3z), what is the partial fraction
decomposition of G(z)“?
31 What function g(n) of the positive ~integer n satisfies the recurrence
x g(d) cp(n/d) = 1,
d\n
where cp is Euler’s totient function?
32 An arithmetic progression is an infinite set of integers
{an+b} = {b,a+b,2a+b,3a+b ,... }.
A set of arithmetic progressions {al n + bl}, . . . , {amn + b,} is called an
exact cover if every nonnegative integer occurs in one and only one of the
progressions. For example, the three progressions {2n}, {4n + l}, (4n + 3)
constitute an exact cover. Show that if {al n + br}, . . , {amn + b,} is an
exact cover such that 2 6 al 6 .. . < a,,,, then a,-1 = a,. Hint: Use
generating functions.
Exam problems
33 What is [w”zn] (ln(1 + z))/(l - wz)?
34 Find a closed form for the generating function tn30 Gn(z)wn, if
(Here m is a fixed positive integer.)
35 Evaluate the sum xO