Concrete mathematics : a foundation for computer science

340 GENERATING FUNCTIONS
F(z)' = i x (n + 1 )(;!F,+r -F,,)2- ; x F,+I .zn ,
7x30 Tl30
and we have the answer we seek:
if
k=O
FkFn-k = 2nF,+ 1--(n+l)F,
5
(7.60)
For example, when n = 3 this formula gives F,JF~ + FlF2 + FzFl + F~F,J =
0+ 1 +1 +0 =2 on the left and (6F4 -4F3)/5 � = (18-8)/5 =2 on the right.
Example 2: Harmonic convolutions.
The efficiency of a certain computer method called “samplesort” depends
on the value of the sum
integers m,n 3 0.
Exercise 5.58 obtains the value of this sum by a somewhat intricate double
induction, using summation factors. It’s much easier to realize that Tm,n is
just the nth term in the convolution of ((i), (A), (i), . . .) with (0, $, i, . . .).
Both sequences have simple generating functions in Table 321:
Therefore, by (7.43),
T
zm
zn = --.
(1 -z)nl+l '
m
xg = ln&.
n>O
1 1 1
m,n = [z”l (, _“,,,,l In 1-z = 'z"-"' (1 -Z)m+l I n -
= U-k’-LJ nnm .
( >
In fact, there are many more sums that boil down to this same sort of
convolution, because we have
1 1
(1 -z)'+s+2 In-
for all T and s. Equating coefficients of 2” gives the general identity
; (‘:“) (s+nl;k)IH,+d-&)
= (r+s+n,L+l)(H.+s+n+~ -H,+,+I) (7.61)