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Recent questions tagged generating-functions

7 votes
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31
GATE Overflow Test Series | Discrete Mathematics | Test 2 | Question: 25
The closed form solution for the $n^{th}$ term $(n > 1)$ of the sequence with generating function $\frac{9x}{1-2x}+ \frac{x}{1-x}$ is given by $1.2^n+1$ $2.9^{n-1} + 1$ $9.2^n + 1$ none of these
The closed form solution for the $n^{th}$ term $(n 1)$ of the sequence with generating function $\frac{9x}{1-2x}+ \frac{x}{1-x}$ is given by$1.2^n+1$$2.9^{n-1} + 1$$9.2^...
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8 votes
1 answer
32
GATE Overflow Test Series | Discrete Mathematics | Test 2 | Question: 28
The $n^{th}$ term of the sequence generated by the function $\frac{2}{(1-x)^2} \cdot \frac{x}{1-x-x^2}$ is represented as $a_n$. The value of $a_6$ is$\_\_\_\_\_\_$.
The $n^{th}$ term of the sequence generated by the function $\frac{2}{(1-x)^2} \cdot \frac{x}{1-x-x^2}$ is represented as $a_n$. The value of $a_6$ is$\_\_\_\_\_\_$.
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3 votes
1 answer
35
Rosen 7e, Advance Counting techniques , Question 6.f
Find the generating function for the sequence $\left \{ a_n \right \} where $ $a_n = \Large \binom{10}{n+1} $ ... $\Large \color{red}{ \frac{( 1+x )^{10} - 1}{x} }$ Please verify
Find the generating function for the sequence $\left \{ a_n \right \} where $$a_n = \Large \binom{10}{n+1} $ for n = 0,1,2,….Sol. $\Large \binom{10}{1} + \binom{10}{2...
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1 votes
0 answers
38
Kenneth Rosen Edition 6th Exercise 7.4 Question 7,8 (Page No. 496)
For each of these generating functions, provide a closed formula for the sequence it determines. $a) (3x − 4)^{3}$ $b) (x^{3} + 1)^{3}$
For each of these generating functions, provide a closed formula for the sequence it determines.$a) (3x − 4)^{3}$$b) (x^{3} + 1)^{3}$
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1 votes
0 answers
41
Generating functions
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 __________ .
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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0 votes
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42
SELF DOUBT GENERATING FUNCTION
Difference between getting closed form of generating function and closed form of the given sequence ,pls someone explain with an example
Difference between getting closed form of generating function and closed form of the given sequence ,pls someone explain with an example
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43
Kenneth_Rosen_GF
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 $a_n=\frac{1}{n+1}-\frac{1}{n+2}$ ... $\sum_{n=0}^{\infty}\frac{1}{(n+2)}.\frac{x^n}{n!}$
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=\f...
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1 votes
3 answers
45
Generating Function
What will be solution of this function for coefficient of $x^{100}$? $\frac{1}{\left ( 1-x^{10} \right )(1-x^{20})(1-x^{50})}$
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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0 votes
1 answer
46
GENERATING FUNCTIONS
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1 answer
49
Kenneth Rosen Edition 6th Exercise 7.4 Question 3e (Page No. 496)
Find closed form for the generating function of the following sequence $\binom{7}{0}, \binom{7}{1}, \binom{7}{2}, ......., \binom{7}{7},0,0,0,0,0,...$
Find closed form for the generating function of the following sequence$\binom{7}{0}, \binom{7}{1}, \binom{7}{2}, ......., \binom{7}{7},0,0,0,0,0,...$
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1 votes
1 answer
50
Self doubt #Generating functions
$\sum_{.}^{.} (3r^{2} + 5r -21) x^{r}$will be equal to :?
$\sum_{.}^{.} (3r^{2} + 5r -21) x^{r}$will be equal to :?
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1 votes
1 answer
51
Kenneth Rosen Edition 6th Exercise 7.4 Question 6 (Page No. 496)
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 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 $n=0,1,2....$
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1 votes
1 answer
52
Kenneth Rosen Edition 7 Exercise 8.4 Question 10 (Page No. 549 )
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) $(1+x+x^2)^3$
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) $(1+x+x^2)^3$
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2 votes
2 answers
53
Kenneth Rosen Edition 6th Exercise 6.4 Question 7g (Page No. 440)
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
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 app...
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2 votes
2 answers
54
Rolling of a dice
The number of ways to roll 5 six sided dice to get sum of 25 is ________. _________________________________________________________ if solving with generating function, then why dividing by $\left ( 1-x \right )$ if equation is $\frac{x\left ( 1-x^{6} \right )}{\left ( 1-x \right )}$?
The number of ways to roll 5 six sided dice to get sum of 25 is ________._________________________________________________________if solving with generating function, the...
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1 votes
1 answer
55
Kenneth Rosen Edition 6th Exercise 6.4 Example 15 (Page No. 435 )
How to find the coefficient ( for eg $x^7$ ) in the generating function$(1+x+x^2+x^3+..)(1+x^2+x^4+x^6+..)(1+x^5+x^{10}+x^{15}+..)$ ?
How to find the coefficient ( for eg $x^7$ ) in the generating function$(1+x+x^2+x^3+..)(1+x^2+x^4+x^6+..)(1+x^5+x^{10}+x^{15}+..)$ ?
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