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COS 341 Discrete Mathematics
Generating Functions
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Administrative Issues• Homework 1 has been graded• Median score: 77
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Generating functions
2
0
0 1( , , , ) : sequence of real numbersof this sequence is
the power serie
Gene
s
rating function
( ) iiia a
a
x x
a a
∞
== ⋅∑
…
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Operations on power series• Addition
• Multiplication by fixed real number
• Shifting the sequence to the right
• Shifting to the left
0 0 1 1 has generating function( , , ) ( ) ( ) a b a b a x b x+ + +…
0 1 has generating functi( , , ) (on )a a a xα α α…
0 1 has generating fu(0, 0 ncti, , on , ) ( )n
n
a a x a x×
… …
1
01 has generating f
(unction
)( , , )
k iii
k k n
a x a xa a
x
−
=+
− ⋅∑…
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• Substituting αx for x
• Substitute xn for x
0 1 21 1
has ge( ,0, 0, nerati,0, 0, ng function ) ( )nn n
a a a a x−× −×
… … …
20 1 2 has generating funct( , , ion) ( ) a a a a xα α α…
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• Differentiation
• Integration
• Multiplication of generating functions
1 2 3 has generating function ( , 2 ,3 ) ( ) (or )'( )da a a a x a xdx
…
1 10 1 22 3
0
has generating function (0, , , ) ( )x
a a a f t dt∫…
( )( ) ( )0 0 0
0
n n nn n nn n n
nn k n kk
a x b x c x
c a b
∞ ∞ ∞
= = =
−=
⋅ ⋅ = ⋅
= ⋅
∑ ∑ ∑∑
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Applying the toolkit2 2 2
2
What is the generating function for (1 ,2 ,th 3 ,e seque(
nce ))
1ka k= +
…
2 3 41 11
x x x xx= + + + + +
−2 3
2
1 1 1 2 3 4(1 ) 1
d x x xx dx x
= = + + + + − −
2 33 2
2 1 1 2 3 2 4 3 5.4(1 ) (1 )
d x x xx dx x
= = ⋅ + ⋅ + ⋅ + + − −
2 33 2
2 1 1 1 2 2 3 3 4 4(1 ) (1 )
x x xx x
− = ⋅ + ⋅ + ⋅ + ⋅ +− −
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An alternate derivation:Generalized Binomial Theorem
2 3r r r r
0 1 2 3(1 )r x x xx + + + +
+ = …
( 1)( 2) ( 1)
!
r r r r r k
k k
− − − +=
…
1 1( 1) ( 1)
1k k
n n k n k
k k n
− + − + −= − = −
−
( 1)( 2) ( 1)
!
( 1)( 2) ( 1)( 1)
!
k
n n n n n k
k k
n n n n k
k
− − − − − − − − +=
+ + + −−=
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An alternate derivation:Generalized Binomial Theorem
2 3r r r r
0 1 2 3(1 )r x x xx + + + +
+ = …
1 1( 1) ( 1)
1k k
n n k n k
k k n
− + − + −= − = −
−
0 0
-1 -1(1 ) ( 1) ( 1)
1n k k k k
k k
n k n kx x x
k n
∞−
= =
∞ + ++ = − = −
−
∑ ∑
0 0
-1 -1(1 ) ( 1) )
1 1(n k k k
k k
n k n kx x x
n n−
= =
∞ ∞+ +− = − =
− −
− ∑ ∑
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An alternate derivation:Generalized Binomial Theorem
0 0
-1 -1(1 ) ( 1) )
1 1(n k k k
k k
n k n kx x x
n n
∞ ∞−
= =
+ +− = − =
− −
− ∑ ∑1
0 0
(1 )0
k k
k k
kx x x−
= =
∞ ∞
− = = ∑ ∑
2
0 0
1(1 ) ( 1)
1k k
k k
kx x k x−
= =
∞ ∞+− +
= = ∑ ∑
3
0 0
2(1 )
2( 2)( 1)
2k k
k k
kx x x
k k−
= =
∞ ∞+−
+ += = ∑ ∑
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An alternate derivation:Generalized Binomial Theorem
2
0 0
1(1 ) ( 1)
1k k
k k
kx x k x−
= =
∞ ∞+− +
= = ∑ ∑
3
0 0
2(1 )
2( 2)( 1)
2k k
k k
kx x x
k k−
= =
∞ ∞+−
+ += = ∑ ∑
3
0
22(1 ) ( 1)( 1)(1 ) k
k
x k k xx−
=
−∞
− + +− − =∑
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More toolkit examples
1 1 12 3 4
of theWhat is the g sequence(1,
enerating functio,
n, , )?
2 31 1 12 3 4
ln(1 1- ) x xxx
x+ + +− = +
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More toolkit examples2 3 41 1
1x x x x
x= + + + + +
−
2 3 4
0 0
(1 )1
x xdt t t t t dtt= + + + + +
−∫ ∫2 3 41 1 1
2 3 4ln(1 ) ln(1) x x xx x + + +− − + = +
2 31 1 12 3 4
ln(1 ) 1x
x x xx + + +− − = +
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Applications to counting
A box contains 30 red, 40 blue and 50 green balls.Balls of the same color are indistinguishable.
How many ways are there of selecting a collection of 70 balls from the box ?
2 30
2 4
2 5
0
7
0
0coefficient (1
of in
))
(
)
1(1 xx x
x
x
x x xxx
× +
+ +
+ ++
+×
++ +
++
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Enter generating functions
2 3
70
2 50 40 2 0
coefficient of i(1 )( (1 )
n1 )x x x xx x xx
xx+ + ++ + + + + ++ ++
312 30 1(1 )
1xx x xx
−+ + + + =−
Sum of first n terms of a geometric series
2
3131 32 33
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1
1
x xx
xx x x
x
= + + +−
= + + +−
Alternately
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Enter generating functions
2 3
70
2 50 40 2 0
coefficient of i(1 )( (1 )
n1 )x x x xx x xx
xx+ + ++ + + + + ++ ++
741
05131
coefficient of i ( (1 )(1 )n 11
1 1
)xx
x xx
xx
−−
−−
−−
31 41 513
31 41 51
0
1(1 )(1 )(1 )
(1 )
2(1 )
2k
k
x x xx
kx x x x
=
∞
− − −−
+= − − − + ∑
70 2 70 31 2 70 41 2 70 51 21061
2 2 2 2+ − + − + − +
− − − =
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More tricks with generating functions
0( ) iiia x a x∞
== ⋅∑
0 nn iib a
==∑
0( )What s i i
iib x b x∞
== ⋅∑
( ) ( ) 1a xb x
x=
−
( )2 20 1 20
(1 )iiib x a a x a x x x
∞
=⋅ = + + + + + +∑
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More tricks with generating functions2 2 2W (1 +2hat )is ?n+
2 33 2
2 1 1 1 2 2 3 3 4 4(1 ) (1 )
x x xx x
− = ⋅ + ⋅ + ⋅ + ⋅ +− −
0 nn iib a
==∑ ( ) ( )
1a xb x
x=
−2
0
2 21
2 2 22
1
1 2
1 2 3
bbb
== += + +
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More tricks with generating functions2 2 2W (1 +2hat )is ?n+
2 33 2
2 1 1 1 2 2 3 3 4 4(1 ) (1 )
x x xx x
− = ⋅ + ⋅ + ⋅ + ⋅ +− −
4 30
2 2 2
2 1(1 ) (1 )
1 2 ( 1)
nn
n
n
b xx x
b n=
∞− =
− −= + + + +
∑
3 22
3 2n
n nb
+ + = −
1
2 12
3 2n
n nb −
+ + = −
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More tricks with generating functions2 2 2W (1 +2hat )is ?n+
2 2 21 1 2
2 12
3 22( 2)( 1) ( 1)
6 2(2 1)( 1)
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nb nn n
n n n n n
n n n
− = + + + + + = − + + += −
+ +=
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More tricks with generating functions
0
(-1What is ?)m
k
k
nk=
∑The generating function for the sequence
( 1 ( ) () is 1 ) kk
nna a x
kx
= − = −
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The generating function for the sequence
is ( ) (1 )1
m nm kk
a xc a xx
−=
= = −−∑
1 (coefficient of 1) =m
m mcm
xn
= − −