# Tagged Questions

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### Use generating functions to prove Pascal’s identity

How do I prove Pascal's identity using generating functions? $C(n,r) = C(n−1,r) + C(n−1,r−1)$ when $n$ and $r$ are positive integers with $r < n$. I am given the hint to use the identity ...
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### Find a closed form for the generating function for this sequence

The sequence: $0, 0, 0, 1, 1, 1, 1, 1, 1, \ldots$ The book gives the answer of $\frac{x^3}{1-x}$ but I'm not sure how to get this answer. I understand the generating function of this sequence will be ...
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### Closed form of $\sum_{k=0}^nk\binom{k}{3}\binom{2n}{k}$

Recently, I came across the following exercise on the course of discrete math Find a closed form for $\sum_{k=0}^nk\binom{k}{3}\binom{2n}{k}$ So I tried some of the usual techniques: Let ...
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### How many ways you can make change for an amount N using A and B monets.

I encountered a quite interesting problem. The question is: How many ways you can make change for an amount N using monets of value A and B, knowing that GCD(A,B)=1. Any idea how to solve this? It ...
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### Find a generating function for $\sum_{k=0}^{n} k^2$

Find a generating function for $\sum_{k=0}^{n} k^2$ I know my solution is wrong, but why? My solution: If $F(x)$ generates $\sum_{k=0}^{n} k^2$ then $F(x)(1-x)$ generates $k^2$. ...
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### Expressing the generating function defined by $b_n = \sum_{k=0}^{n} 3^k\cdot a_k$

The title is probably somewhat unclear, sorry if it is.. Let $F$ be the generating function of the sequence $(a_n)_{n=0}^{\infty}$ Use $F$ to express the generating function for ...
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### Calculate the $26$ term for the generating function.

Let $\lambda x.(1+x+x^{10})^{20}$. What is the the $26$ term for the series generated by this function? Thanks.
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### About the generating function of the sum of roll dice values.

I thought this exercise would be fairly easy, but it seems i can't find a proper approach to it. I have to prove that $f(x) = (1-x-x^2-x^3-x^4-x^5-x^6)^{-1}$ is the function that generates the number ...
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### Find generating function of given problem?

please help me to find the generating function of this problem $a_k = ( k + 1) for k=0,1,2,3,...$
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### Generating Functions in Discrete Mathematics in Computer Science

Hey Guys can anyone help me with the following question in Discrete Structures in Mathematics, relating generating functions Find a closed form for the generating function for the sequence $\{a_n\}$, ...
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### Generating function question

let say I have $90 balls$ $S_{1} =$ green balls $S_{2} =$ orange balls $S_{3} =$ red balls I have the following limitations: No limit Choose at most 60 balls from each color choose even ...
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### How to obtain asymptotic form for sequence, given generating function

Let $a_n$ be the number of ways to obtain the amount of $n$ cents, using a supply of 1-cent coins, 3 types of 2-cent coins, and 4-cent coins. Then, $a_n$ is the coefficient of $x^n$ in ...
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### Generating functions homework question 3

This is a Homework question Determine the Closed Form generating function for the sequence $a_0,a_1,a_2...,$ where $a_n$ is the number of partitions of the non negative integer n into a) even ...
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### Generating Functions Homework Question 2

This is a HW question The question is to use generating functions to count the number of six digit (positive) integers whose digits sum to $42$. Ex. $978468$ is a six digit integer whose digits sum ...
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### generating function Homework Question 1

This is a HW question I am asked to find a closed form generating function for $1,1,0,1,1,0,1,1,0....$ so then $f(x)=x^0+x^1+0x^2+x^3+x^4+0x^5+x^6+x^7+0x^8$ could use some hint or help.
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### Recurrence relations and generating functions question

Let $A_n$ be the set of different paving of a $2\times n$ using $2\times 1$ or $1 \times 2$ tiles. We'll define $a_n$=$|A_n|$. 1] Find recurrence relation: I found it -> $a_n=a_{n-1}+a_{n-2}$ with ...
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### A small question about generating functions

just a small question: When should I use infinite geometric sequence and when should I use finite geometric sequence when solving problems involving combinations? For instance, for the problem: How ...
Let $P$ denote the set of primes and let $s\in\{-1,1\}$. How can you interpret the coefficient of $x^n$ in the power series expansion of $$\prod_{p\in P} (1+sx^p)^s$$ for either choice of $s$? I ...
In how many ways can you choose $10$ balls, of a pile of balls containing $10$ identical blue balls, $5$ identical green balls and $5$ identical red balls? My solution (not sure if correct, would ...