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3. We found in class that if Cn counts the number of ways to make change (in standard coins) for n cents, the generating function for this sequence is 8 Σ cnx" n=0 = 1 (1 − x)(1 − x5) (1 − x¹⁰) (1 – x²5) · - Let an again be the number of ways to make change for n cents, but with the added feature that we care about the order in which the coins are chosen. We could say, for example, that an is the number of ways to feed coins totalling n cents one by one into a machine. Give a closed-form generating function for the sequence (an). [Hint: you will use 0 coins in total, or 1 coin in total, or 2 coins in total...]

3. We found in class that if Cn counts the number of ways to make change (in standard coins) for n cents, the generating function for this sequence is 8 Σ cnx" n=0 = 1 (1 − x)(1 − x5) (1 − x¹⁰) (1 – x²5) · - Let an again be the number of ways to make change for n cents, but with the added feature that we care about the order in which the coins are chosen. We could say, for example, that an is the number of ways to feed coins totalling n cents one by one into a machine. Give a closed-form generating function for the sequence (an). [Hint: you will use 0 coins in total, or 1 coin in total, or 2 coins in total...]

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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3. We found in class that if Cn counts the number of ways to make change (in standard coins) for n cents, the generating function for this sequence is Σ n=0 Cnxn = 1 (1 − x) (1 − x5) (1 – x¹0) (1 − x²5) - - Let an again be the number of ways to make change for n cents, but with the added feature that we care about the order in which the coins are chosen. We could say, for example, that an is the number of ways to feed coins totalling n cents one by one into a machine. Give a closed-form generating function for the sequence (an). [Hint: you will use 0 coins in total, or 1 coin in total, or 2 coins in total...]
Transcribed Image Text:3. We found in class that if Cn counts the number of ways to make change (in standard coins) for n cents, the generating function for this sequence is Σ n=0 Cnxn = 1 (1 − x) (1 − x5) (1 – x¹0) (1 − x²5) - - Let an again be the number of ways to make change for n cents, but with the added feature that we care about the order in which the coins are chosen. We could say, for example, that an is the number of ways to feed coins totalling n cents one by one into a machine. Give a closed-form generating function for the sequence (an). [Hint: you will use 0 coins in total, or 1 coin in total, or 2 coins in total...]
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