search
Log In

Recent questions tagged generating-functions

3 votes
1 answer
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 $\frac{1}{n+1}\\$ $\frac{1}{n+2}\\$ $\frac{1}{n(n+1)}\\$ $\frac{1}{(n+1)(n+2)}$
asked May 11, 2019 in Combinatory akash.dinkar12 706 views
0 votes
2 answers
2
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 $a)3\left ( 1+3x \right )^{-2}$ $b)3\left ( 1-3x \right )^{-2}$ $c)2\left ( 1+3x \right )^{-3}$ $d)2\left ( 1-3x \right )^{-3}$
asked Mar 8, 2019 in Combinatory srestha 388 views
3 votes
1 answer
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
asked Mar 7, 2019 in Combinatory Mk Utkarsh 160 views
1 vote
0 answers
5
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.
asked Jan 13, 2019 in Combinatory jhaanuj2108 173 views
1 vote
0 answers
6
2 votes
1 answer
7
Which one of the following best expresses the generating function sequence $\{a_n\}$, for the given closed form representation? $F(x) = \frac{1}{1-x-x^2}$ $a_n=a_{n-1}+3, n>0, a_0=1$ $a_n=a_{n-1}+a_{n-2}, n>1, a_0=1, a_1=1$ $a_n=2n+3, n>1$ $a_n=2a_{n-1}+3, n>1, a_0=1$
asked Dec 27, 2018 in Set Theory & Algebra Ruturaj Mohanty 548 views
6 votes
2 answers
8
1 vote
0 answers
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 __________ .
asked Dec 12, 2018 in Mathematical Logic shraddha priya 250 views
0 votes
0 answers
10
Difference between getting closed form of generating function and closed form of the given sequence ,pls someone explain with an example
asked Dec 10, 2018 in Combinatory codingo1234 97 views
0 votes
0 answers
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 $a_n=\frac{1}{n+1}-\frac{1}{n+2}$ ... $\sum_{n=0}^{\infty}\frac{1}{(n+2)}.\frac{x^n}{n!}$
asked Dec 4, 2018 in Combinatory Ayush Upadhyaya 123 views
3 votes
1 answer
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
asked Dec 4, 2018 in Combinatory srestha 562 views
1 vote
0 answers
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}}) $
asked Dec 3, 2018 in Combinatory !KARAN 385 views
1 vote
3 answers
14
What will be solution of this function for coefficient of $x^{100}$? $\frac{1}{\left ( 1-x^{10} \right )(1-x^{20})(1-x^{50})}$
asked Dec 3, 2018 in Combinatory srestha 589 views
0 votes
1 answer
15
0 votes
0 answers
16
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 ?
asked Oct 16, 2018 in Combinatory tonystark 128 views
...