# Recent questions tagged generating-functions

1
Let $C_i(i=0,1,2...n)$ be the coefficient of $x^i$ in $(1+x)^n$.Then $\frac{C_0}{2} – \frac{C_1}{3}+\frac{C_2}{4}-\dots +(-1)^n \frac{C_n}{n+2}$ is equal to $\frac{1}{n+1}\\$ $\frac{1}{n+2}\\$ $\frac{1}{n(n+1)}\\$ $\frac{1}{(n+1)(n+2)}$
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The generating function of the sequence $\left \{ a_{0},a_{1},a_{2}..........a_{n}………...\infty \right \}$ where $a_{n}=\left ( n+2 \right )\left ( n+1 \right ).3^{n}$ is $a)3\left ( 1+3x \right )^{-2}$ $b)3\left ( 1-3x \right )^{-2}$ $c)2\left ( 1+3x \right )^{-3}$ $d)2\left ( 1-3x \right )^{-3}$
3
Find the generating function for the sequence $\left \{ a_n \right \} where$ $a_n = \Large \binom{10}{n+1}$ for n = 0,1,2, . Sol. $\Large \binom{10}{1} + \binom{10}{2}x + \binom{10}{3}x^2 + \binom{10}{4} x^3 + .... + \binom{10}{10}x^{9}$ multiplying and dividing above equation ... , $\Large - \frac{1}{x} + \frac{1}{x} ( 1+x)^{10}$ $\Large \color{red}{ \frac{( 1+x )^{10} - 1}{x} }$ Please verify
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Let We define Then ar is equal to. $\binom{r}{2019}$ $\binom{r}{r + 2018}$ $\binom{r}{2019 – r}$ $\binom{r}{r – 2018}$ Can anyone tell me if this type of question is in Gate 2019 syllabus or not because I have never seen such question in previous year question? If yes, then when can I learn this stuff from. Because I am unable to understand the whole solution.
1 vote
6
For each of these generating functions, provide a closed formula for the sequence it determines. $a) (3x − 4)^{3}$ $b) (x^{3} + 1)^{3}$
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Which one of the following best expresses the generating function sequence $\{a_n\}$, for the given closed form representation? $F(x) = \frac{1}{1-x-x^2}$ $a_n=a_{n-1}+3, n>0, a_0=1$ $a_n=a_{n-1}+a_{n-2}, n>1, a_0=1, a_1=1$ $a_n=2n+3, n>1$ $a_n=2a_{n-1}+3, n>1, a_0=1$
8
Let $M(x) = \frac{x^{2018}}{(1-x)^{2019}}$ we define $M(x) = \sum_{r=0}^{\infty}a_{r}x^{r}$ ,then $a_{r}$ is equal to- $A)\binom{r}{2019}$ $B)\binom{r}{r+2018}$ $C)\binom{r}{2019-r}$ $D)\binom{r}{r-2018}$
1 vote
9
The number of ways can 10 balls be chosen from an urn containing 10 identical green balls, 5 identical yellow balls and 3 identical blue balls are __________ .
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Difference between getting closed form of generating function and closed form of the given sequence ,pls someone explain with an example
11
Find a closed form for the exponential generating function for the sequence $\{a_n\}$ where $a_n=\frac{1}{(n+1)(n+2)}$ I broke it down into partial fractions and got $a_n=\frac{1}{n+1}-\frac{1}{n+2}$ ... $\sum_{n=0}^{\infty}\frac{1}{(n+2)}.\frac{x^n}{n!}$
12
Use Generating function to determine,the number of different ways $10$ identical balloons can be given to four children if each child receives atleast $2$ ballons? Ans given $(x^{2}+x^{3}+.........................)^{4}$ But as there is a upper bound I think answer will be $(x^{2}+x^{3}+.........................+{\color{Red} {x^{10}}})^{4}$ Which one is correct? plz confirm
1 vote
13
How to find coefficient of $x^{100}$? $= (1+x^{10}+(x^{10})^2 + \dots)(1+x^{20}+(x^{20})^2 + \dots)(1+x^{50}+(x^{50})^2 + \dots)\\ = (\frac{1}{1-x^{10}}).(\frac{1}{1-x^{20}}).(\frac{1}{1-x^{50}})$
1 vote
14
What will be solution of this function for coefficient of $x^{100}$? $\frac{1}{\left ( 1-x^{10} \right )(1-x^{20})(1-x^{50})}$
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Given only a Generating Function in closed form can we find the sequence it represents? If so, how? Please explain with an example.If not, given a GF(closed form) in general what information does it convey to us about a series if the series is not provided ?
1 vote
17
Find a closed form of the generating function of the following sequence $0, 1, -2, 4, -8, 16, -32,64,.....$
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Find closed form for the generating function of the following sequence $\binom{7}{0}, \binom{7}{1}, \binom{7}{2}, ......., \binom{7}{7},0,0,0,0,0,...$
1 vote
19
$\sum_{.}^{.} (3r^{2} + 5r -21) x^{r}$will be equal to :?
1 vote
Find the closed form for the generating function for the sequence $\{a_n\}$ where (a)$a_n=\binom{n}{2}$ for $n=0,1,2....$ (b)$a_n=\binom{10}{n+1}$ for $n=0,1,2....$
Find the coefficient of $x^9$ in the power series of each of these functions. a) $(x^3+x^5+x^6).(x^3+x^4).(x+x^2+x^3+x^4+⋯)$ b) $(1+x+x^2)^3$
Please tell me the approach of solving the question for finding a closed formula for the given generating function:- $x^{2} / (1-x)^{2}$ . Please determine the general approach how to solve