Concrete mathematics : a foundation for computer science

Hint: 1f the sequence
consists
of binomial coeficients,
its generating
function usually
involves a binomial,
1+z.
Table 321 Simple sequences and their generating functions.
7.2 BASIC MANEUVERS 321
sequence generating function closed form
(1 , o,o, 0, o,o,. . )
(0,. . . I O,l,O,O ,... 1)
(l,l,l,l,l,l,...)
(1,-1,1,-1,1,-l,...)
(l,O, l,O, l,O,. . . )
(1,0,...,0,1,0,....0,1,0,
(1,43,4,5,6,...)
(1,2,4,8,16,32,...)
(1,4,6,4,1,0,0,...)
(k(;),(;),...)
(Lc,(':'),(':') ,...)
(l,c,cQ3,...)
(1, (mm+'), (mm+2), ("Z3),
(o,L;>;,$,...)
(OJ-;,;,-;,...)
( 11'111
) ‘2’6’24’,20”” >
)
x
,>o[n=Ol Zn
fIoLn=ml Zn
t ’ zn
n30
tn>Op 1” zn
tn>O [AnI 9
/
tn>O [m\nlC
,
xn>o (n + 1) zn
t n>O 2” =n
xn:O (
4
n
) zn
c n
t..-.( )
EnI (":"j zn
tn>O
n n
> Loi z
m+n
t iz:
n2l n
ix
n31
t 1%
7x20 n!
) zn
(-v+’ Zn
1
zm
1
1-Z
1
l+z
1
l-22
1
l-zm
1
(1 - 2)2
1
l-22
(1 + 2J4
(1 + zy
1
(1 - z)C
1
l-cz
1
(1 - z)m+'
In
1
- 1-Z
ln(1 + 2)
them can be derived quickly from the others by using the basic operations of
Table 320; therefore the memory work isn’t very hard.
For example, let’s consider the sequence (1,2,3,4, . ), whose generating
function l/( 1 - z)~ is often useful. This generating function appears near the
eL