generating functions - Number of ordered, unlabeled binary rooted trees with n nodes and k leafs

I want to find the number of ordered, unlabeled binary rooted trees with $n$ nodes and $k$ leafs as an exercise.To be more precise. I am interested in objects like this ((c) 2015 M. Fulmek, PS Kombinatorik) where the line below $n=4$ reads “… and the 7 trees above vertically filpped” Assigning the weight $w(x)W = z^n y^k$ to every rooted tree $W$ with $n$ nodes and $k$ leafs. The generating function in two indeterminates should start like this$$ T(z, y) = \sum_W w(x)W = zy + z^2 2y + z^3\left(y^2 + 4y\right) + z^4\left(6y^2 + 8y\right) + \ldot...Read more

Find the generating function

I'm trying to find the generating function for this sequence: $$0,0,3,0,9,17,33,65,129,257...$$What I know so far:$$0\cdot x^0 + 0\cdot x^1 + 3\cdot x^2 + 0 \cdot x^3$$and$$x^5(9+17x+33x^2+65x^3+129^4+257^5....$$So what I have to do now Is just add the two, but I'm not 100% sure how to give a closed form answer for the second sequence. From what I can see it has an arithmetic progression of $8\cdot n$, so I've gotten this:$$3x^2+x^5\sum_{n=1}^∞(9+8n)$$What else do I need to do ? have I found the generating function?...Read more

recursion - Problem solving recurrences with generating functions

I'm trying to solve this linear recurrence with generating functions, but I keep getting stuck on the last few steps. I found the generating function, but after splitting it into partial fractions and putting it in sigma notation, I don't know how to simplify it into one summation:$g_0$=1, $g_1$=2, $g_2$=-2$g_n$=7$g_{n-1}$-16$g_{n-2}$+12$g_{n-3}$I got up to the point;$$\frac{1-5z}{1-7z+16z^2-12z^3}=\frac{7-8z}{(1-2z)^2}+\frac{-6}{1-3z}$$and finally;$$=(7-8z)\sum_{n=0}^\infty(n+1)2^nz^n-6\sum_{n=0}^\infty3^nz^n$$I have no clue what to do now, bu...Read more

Convert a closed-form generating function to a sequence

I need some help with an assignment question:I must determine the sequence generated by the following generating function:$2x^3 \over 1 - 5x ^ 2 $In class we have only gone from the sequence to the closed form so, I am not really sure how to begin on this. I feel like I should begin by separating the functions so that we are working with $ 2x^3 \cdot {1 \over 1-5x^2}$ which is closer to the form that I am used to seeing come from the sequence. We would get these closed-form functions out from the sequence using a table, so logically going the o...Read more

Closed form generating function

Q. What is the generating function for the sequence 1,1,1,1,1,1?Ans. The generating function for the sequence is $1+x+x^2+x^3+x^4+x^5.$Now we have** $\frac{(x^6-1)}{(x-1)} = 1+x+x^2+x^3+x^4+x^5$.**Consequently, $G(x) = \frac{x^6-1}{x-1}$ is the required generating function.I don't understand the line closed by **. What method has been applied to get that....Read more

find closed form for recurrence relation using generating function

I have this recurrence relation:$$\begin{equation} \begin{cases} a_n = 2a_{n-1} + n & (n\geq1)\\ a_0 = 1 \end{cases}\end{equation}$$Set:$$f(x) = \sum_{n=0}^\infty a_nx^n$$I solved in this way:$$\sum_{n=1}^\infty a_nx^n = 2\sum_{n=1}^\infty a_{n-1}x^n + \sum_{n=1}^\infty nx^n$$$$f(x)-1 = \frac{2}{x}\big(f(x)-1\big) + \frac{1}{1-x}-1$$$$f(x)-1 = \frac{2}{x}\big(f(x)-1\big) + \frac{1}{(1-x)^2}-1$$$$f(x)\big(1-\frac{2}{x}\big) = \frac{-2(1-x)^2+x}{x(1-x)^2}$$$$f(x) = \frac{-2x^2+5x-2}{(1-x)^2(x-2)}$$and then I get:$$f(x) = \frac{A}{1-x...Read more

Generating Function, and Combinations

How many ways are there to obtain an even sum when 10 indistinguishable disce are rolled? hint: let $x_i$ be the number of dice showing the number $i$. OK, so the answer is $(\frac{1}{1-z})^3 [\binom{3}{1}(\frac{1}{1-z^2})(\frac{z}{1-z^2})^2+(\frac{1}{1-z^2})^3] $This is the same as saying $x_1+x_2+x_3+x_4+x_5+x_6=10$ with the condition that $x_1+2x_2+3x_3+4x_4+5x_5+6x_6$ is even. This is what the text says. This is what I understand so far$(\frac{1}{1-z})^3$ this is the $2x_2+4x_4+6x_6$ since these will always be even, but in reality they are ...Read more

integer partitions - Generating functions finding $ a_n $ formula

When I have generating function in form $$ A(x) = \frac{11x-1}{(1-3x)(1-7x)} $$I know that the one way to find formula for $ a_n $ is to find partial fraction from this formula and then change this to power series. But when I have to deal with generating function of integer partition I don't know how I should start.I have function:$$ \sum_{n=0}^\infty a_nx^n = (1+x+x^2) \prod_{i=1}^\infty \frac{1}{1-x^{2i+1} }$$And I have to check if it is true for this function that:$a_{100}>20 $ or $a_7 = 5$ or $a_n \neq 0$ for $n = 0,1,2...$I know that $...Read more

How to find a generating function in simple, closed form

I’m stuck on something in generating functionology. The first problem asks:Find the ordinary power series generating functions of the sequence in simple closed form for the sequence $a_n = n$. The sequence is defined as $n ≥ 0$. I figured out how to get to $A(x) = x/((1-x)^2)$. That’s not an issue.However, the book lists the answer as $(xD)(1/(1-x)) = x/((1-x)^2)$Where did the D come from? How can I get my answer in terms of D?...Read more