Question 5. Let an be the number of even sized subsets of {1,...n}. (You already know what this is, but pretend that you don't.) Use Theorem 8.21 from the book to determine the exponential generating function for an. Of course your answer will give an explicit formula for an. Hint: You might need to know the Maclaurin series for cosh(x).
Question 5. Let an be the number of even sized subsets of {1,...n}. (You already know what this is, but pretend that you don't.) Use Theorem 8.21 from the book to determine the exponential generating function for an. Of course your answer will give an explicit formula for an. Hint: You might need to know the Maclaurin series for cosh(x).
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
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Chapter2: Second-order Linear Odes
Section: Chapter Questions
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Transcribed Image Text:Question 5. Let an be the number of even sized subsets of {1,.n}. (You already
know what this is, but pretend that you don't.) Use Theorem 8.21 from the book to
determine the exponential generating function for an.
Of course your answer will give an explicit formula for an.
Hint: You might need to know the Maclaurin series for cosh(x).
![Theorem 8.21 (Product formula, exponential version). Let an be
the number of ways to build a certain structure on an n-element set, and
let bn be the number of way to build another structure on an n-element set.
Let cn be the number of ways to separate [n] into the disjoint subsets S
and T, (S U T = [n]), and then to build a structure of the first kind on
S, and a structure of the second kind on T. Let A(x), B(x), and C(x) be
the respective exrponential generating functions of the sequences {an}, {bn},
and {cn}. Then
A(x)B(x) = C(x).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fc2397270-c13f-4cbc-8085-ef9e58d3496f%2Ff4fd8fb2-3d0b-4f5d-b0bc-b0f7e26be560%2F5g50nei_processed.png&w=3840&q=75)
Transcribed Image Text:Theorem 8.21 (Product formula, exponential version). Let an be
the number of ways to build a certain structure on an n-element set, and
let bn be the number of way to build another structure on an n-element set.
Let cn be the number of ways to separate [n] into the disjoint subsets S
and T, (S U T = [n]), and then to build a structure of the first kind on
S, and a structure of the second kind on T. Let A(x), B(x), and C(x) be
the respective exrponential generating functions of the sequences {an}, {bn},
and {cn}. Then
A(x)B(x) = C(x).
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