Concrete mathematics : a foundation for computer science

358 GENERATING FUNCTIONS
5 Find a generating function S(z) such that
Basics
[z”l S(z) = x (;) (,I,,) .
k
6 Show that the recurrence (7.32) can be solved by the repertoire method,
without using generating functions.
7 Solve the recurrence
40 = 1;
gn = gn I +29,-2+...+ng0,
8 What is [z”] (ln(1 - z))z:‘(l - z)~+‘?
for n > 0.
9 Use the result of the previous exercise to evaluate xE=, HkHnpk.
10 Set r = s = -l/2 in identity (7.61) and then remove all occurrences of
l/2 by using tricks like (5.36). What amazing identity do you deduce? I deduce that Clark
11 This problem, whose three parts are independent, gives practice in the
manipulation of generating functions. We assume that A(z) = x:, anzn,
Kent is really
superman.
B(z) = t, bnzn, C(z) = tncnzn,
negative n.
and that the coefficients are zero for
a If ‘TX = tj+,Zk '
13 Prove Raney’s generalized lemma, which is stated just before (7.6~).
14 Solve the recurrence
go = 0, 91 = 1,
gkgn-k,
by using an exponential generating function.
for n > 1,