Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN: 9781133382119
Author: Swokowski
Publisher: Cengage
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Question
Answer only question six and it's a applied discrete mathematics question
![1.
one has
2.
3.
4.
Let Dn be the number of derangements of [n]. Prove that for n ≥ 1
Dn=nDn−1 + (−1)”.
The generating function of the sequence (an)n is
f(x)=
=
Use it to find a formula for an
X
(1-x)2(13x)
Use the recurrence relation from Problem 1 to find the exponential
generating function for the sequence (Dn)n. Recall that Do = 1.
The following table gives the values of pk (n), the number of integer
partitions of n into exactly k parts. Complete the 8th row and use it to find
p(8), the number of integer partitions of 8.
k
1
3
4
5
6
7
8
n
1
1
0
0
0
0
0
0
2
1
1
0
0
0
0
0
3
1
1
1
0
0
0
0
0
4
1
2
1
0
0
0
0
5
1
2
2
1
1
0
6
1
3
3
2
1
1
0
0
7
1
3
4
3
2
1
1
0
8
5.
Consider the sequence (hn)n defined recursively as follows
6.
ho = 1
h₁
=
2
hn=3hn-1+4hn-2 n ≥ 2.
Find the generating function of this sequence.
Consider the sequence (An)n defined recursively as follows
Ao
=
[An+1 ¦
1
=
n
Ai
Σ
n ≥ 0.
(n - i)!'
i=0
Find the generating function of this sequence.](https://content.bartleby.com/qna-images/question/ed04f254-2b14-4f51-a897-ea8ba8024a7a/57b0f06c-1d64-4e35-a165-e8a273640f2c/c5o25vb_thumbnail.jpeg)
Transcribed Image Text:1.
one has
2.
3.
4.
Let Dn be the number of derangements of [n]. Prove that for n ≥ 1
Dn=nDn−1 + (−1)”.
The generating function of the sequence (an)n is
f(x)=
=
Use it to find a formula for an
X
(1-x)2(13x)
Use the recurrence relation from Problem 1 to find the exponential
generating function for the sequence (Dn)n. Recall that Do = 1.
The following table gives the values of pk (n), the number of integer
partitions of n into exactly k parts. Complete the 8th row and use it to find
p(8), the number of integer partitions of 8.
k
1
3
4
5
6
7
8
n
1
1
0
0
0
0
0
0
2
1
1
0
0
0
0
0
3
1
1
1
0
0
0
0
0
4
1
2
1
0
0
0
0
5
1
2
2
1
1
0
6
1
3
3
2
1
1
0
0
7
1
3
4
3
2
1
1
0
8
5.
Consider the sequence (hn)n defined recursively as follows
6.
ho = 1
h₁
=
2
hn=3hn-1+4hn-2 n ≥ 2.
Find the generating function of this sequence.
Consider the sequence (An)n defined recursively as follows
Ao
=
[An+1 ¦
1
=
n
Ai
Σ
n ≥ 0.
(n - i)!'
i=0
Find the generating function of this sequence.
Expert Solution
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