Tagged Questions

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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Generating function for $\sum_{n=0}^{\infty} n^k x^n$

I would like to get the closed form for this generating function, assuming the $k$ is given up front / held constant: $\sum_{n=0}^{\infty} n^k x^n$ However I don't know if this is too advanced for me ...
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How to formally use Taylor expansions for $n$th derivatives and generating functions?

When deriving Catalan numbers, the generating function takes on this form: $$C(x) = \frac{1}{2} (1 - \sqrt{1-4x}) = \frac{1}{2} (1 - f(x))$$ where $f(x) = \sqrt{1-4x}$ How does one formally show ...
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Deriving sum of powers formula using generating functions

Just for fun I wanted to try to derive a formula for the sum of $p$-powers using generating functions, but without using any literature or websites for help. However I do need a small push or hint. ...
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Does the pgf uniquely determine the distribution? [closed]

I know that the characteristic function of a random variable uniquely determines the distribution, but I'm just curious whether the probability generating function does so too(assuming that it exists)....
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Transforming generating functions into algorithms that generate combinatorial objects

I've stumbled upon this paper where they write about random sampling of combinatorial objects. For sampling to be proper one has to know some core numbers (probabilities). However, I'm not interested ...
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Probability generating function of some “random walk”

Let $S_n=\sum^n_{r=0}X_r$ be a left-continuous random walk on the integers with a retaining barrier at zero. More specifically, we assume that the $X_r$ are identically distributed integer-valued ...
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Is there a formula to multiply many series (More than two) using the Cauchy product? If there isn't, please tell me how I can write this formula $\left( \frac{1}{a-e^x} \right) ^{n+1}$as the ...
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Is $G(\alpha s)/G(\alpha)$ a probability generating function

Suppose $G$ is a probability generating function. Let $\alpha\in [0,1]$, then is $\frac{G(\alpha s)}{G(\alpha)}$ necessarily a probability generating function?
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Deriving the sequence if the generating function is irreducible?

I am trying to better understand generating functions and how they can be derived / manipulated / etc. Right now I am operating on this identity, slightly modified from the answer here: For a ...
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How do I get a sequence from a generating function?

For example if I have the generating function $\frac{1}{1-2x}$ then it corresponds to the sequence $1 + 2x + 4x^2 + 8x^3 +~...$. I know how to start from the sequence and get the generating function, ...
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Finding the generating function of a recurrence relation in dependence of a variable

Given this inhomogeneous linear recurrence relation of 2nd order : $F_n = F_{n-2} + a$ for $n \geq 2$ with $F_1 = 1$ and $F_0 = 0$ How do I find the generating function of this recurrence ...
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Moment Generating Function for $r$th central moment

When using moment generating functions, to find the $n$th raw moment ("$n$th moment about the origin"), you take the $n$th derivative of the MGF and evaluate at $t=0$. To find the $m$th central ...
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How do I compute the generating function for this number sequence?

I am trying to compute the generating function for the number sequence given by $a_n = (-1)^n$. I know that the solution is $A(x) = \frac{1}{1+x}$ but when I try to solve it using the procedure of ...
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Probability of N unrelated events, each with different probabilities, what is the chance X number of outcomes occur

Given the probability of N unrelated events, each with different probabilities, what is the chance X number of outcomes occur? Said specifically there are 8 unrelated contracts, what is the chance a ...
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Given a probability generating function what is the $r$th term
Given that pgf is $G(H(\xi))=\frac{1+\xi}{3-\xi}$. Where $H(\xi)=\frac{1}{2}(1+\xi)$ and $G(\xi) = \frac{\xi}{2-\xi}$. And that $G(H(\xi))$ is the pgf of some random variable $Y$. How does one get the ...