Mathematics for Computer Science

“mcs” — 2017/3/3 — 11:21 — page 696 — #704
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Chapter 16
Generating Functions
(c) 1, 0, 1, 0, 1, 0, 1,. . .
(d) 1, 4, 6, 4, 1, 0, 0, 0,. . .
(e) 1, 2, 3, 4, 5,. . .
(f) 1, 4, 9, 16, 25,. . .
(g) 1, 1, 1/2, 1/6, 1/24, 1/120,. . .
Class Problems
Problem 16.5.
Let A.x/ D P 1
nD0 a nx n . Then it’s easy to check that
a n D A.n/ .0/
;
nŠ
where A .n/ is the nth derivative of A. Use this fact (which you may assume) instead
of the Convolution Counting Principle 16.2.3, to prove that
1
1 !
.1 x/ D X n C k 1
x n :
k k 1
nD0
So if we didn’t already know the Bookkeeper Rule from Section 15.6, we could
have proved it from this calculation and the Convolution Rule for generating functions.
Problem 16.6. (a) Let
S.x/ WWD x2 C x
.1 x/ 3 :
What is the coefficient of x n in the generating function series for S.x/?
(b) Explain why S.x/=.1 x/ is the generating function for the sums of squares.
That is, the coefficient of x n in the series for S.x/=.1 x/ is P n
kD1 k2 .
(c) Use the previous parts to prove that
nX
k 2 D
kD1
n.n C 1/.2n C 1/
:
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