# Questions tagged [generating-functions]

Generating functions are formed by making a series $\sum_{n\geq 0} a_n x^n$ out of a sequence $a_n$. They are used to count objects in enumerative combinatorics.

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### Find the generating function for the number of integer solutions to $x_1 + x_2 + x_3 + x_4 = r$ with $1 \le x_1 \le x_2 \le x_3 \le x_4$

Find the generating function for the number of integer solutions to $x_1 + x_2 + x_3 + x_4 = r$ with $1 \le x_1 \le x_2 \le x_3 \le x_4$. The textbook has showed me the solution, so I do know how to ...
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### for characteristic function $\phi$, prove that $1-|\phi(2u)|^2\le 4( 1-|\phi(u)|^2 )$

I know that, If $X$ and $Y$ are two independent random variables then $$\phi_{X+Y}(u) =\phi_{X}(u)\cdot\phi_{Y}(u)$$ and $\phi_{-X}(u)=\overline{\phi_{X}(u)}$ How to proceed further?any steps to ...
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### Generating functions and algebricity

I study Generating Functions and i see the example 5.1a p.p 9 in paper below. i can't understand the author how solved it? can you help me? Paper: "SOME APPLICATIONS AND TECHNIQUES FOR GENERATING ...
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### Computing coefficients of the generating function counting trees with forbidden members

How can I compute the coefficients $r^{(n)}_p$ with a Python code using the following equations? \label{10} r^{(n)}(x)= S^{(n)}(x)-\frac{1}{2}\left[ (S^{(n)}(x))^2-S^{(n)}(x^2)\right] ...
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### Almost all trees are cospectral (Allen Schwenk's 1973 article)

I am currently working on the following article: https://www.researchgate.net/publication/245264768_Almost_all_trees_are_cospectral. There are a few things that I don't understand, and since the ...
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### For a p.g.f, of $X, f_X(s)$, is $s$ really “just” a dummy variable?

The p.g.f, $f_X(s)$ of a random variable $X$ is given by $$f_X(s) = \sum_{x = 0}^\infty P(X = x)s^x$$ Usually, it's said the $s$ is just a dummy variable, but having been exposed to the result that ...
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### Kids eating candy [closed]

Determine how many ways I can distribute 80 candies to 3 kids, such that: The first kid receives an arbitrary number of candies (possibly 0). The second kid receives an even positive number of ...
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### How to convert $\frac{1}{(1-x)(1-x^3)}$ into a sum of multiple fractions?

What I mean is, that one can convert $\frac{1}{(1-x)^2(1-x^2)}$ into the following sum: $\frac{1}{8}(\frac{1}{1+x}+\frac{1}{1-x} +\frac{2}{(1-x)^2} + \frac{4}{(1-x^3)})$ But I can't seem to do the ...
Let Y~Poisson($\lambda$) be a random variable. Use $G(z)$ to show that: $\mathbb P(Y\geq2\lambda)\leq e^\lambda 2^{-2\lambda}$, where $G$ is the probability generating function of Y, defined as $G(z)=\... 2answers 50 views ### Solve the recursion given by$f(n) = 2f(n-1) + \frac{(-1)^n}{n!}$using generating functions. Assume that$f(0) = 0, f(1) = -1$.$g(x)$is$f(n)$'s generating function, I got to the expression:$ g(x) = \frac{e^{-x}-1}{1-2x} $But am now stuck, since I can't find a power series for the ... 1answer 65 views ### Find the coefficient of$x^{24}$in the power series of$e^{2x}(e^x-1)$. I tried to work through the problem algebraically, and got to$\frac{2^{24}}{24!}(2^{24}-1)$, but comparing with the Taylor series generated for the function by Wolfram Alpha, it seems to be incorrect.... 1answer 63 views ### Number of non-negative integer solutions for$x+y+z = n^2$What is the number of solutions for$x+y+z = n^2$for$x,y,z$non-negative integers? I thought to use generating functions. I know that the generating function for$x_1+x_2+...+x_k= n$when$x_i \in ...
How many lattice paths in $\mathbb{Z}$ of length $n$ with steps in $\{-1,0,+1\}$ visit $m$ distinct points? Notice that this is just the number of lattice paths $P$ such that $\max P - \min P + 1 = m$...