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.
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.
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter5: Inverse, Exponential, And Logarithmic Functions
Section5.6: Exponential And Logarithmic Equations
Problem 64E
Related questions
Question
Answer only question three 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.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fed04f254-2b14-4f51-a897-ea8ba8024a7a%2F49c09446-1a19-419a-94d6-4f9380dc8164%2Ffl8bfmc_processed.jpeg&w=3840&q=75)
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.
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