# 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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### generating function of recursion with a hyperbolic function

I am looking to solve a recursion for a certain sequence $\{a_n\}_{n \geq 1}$ through its generating function $$f(x)=\sum_{n\geq 1} a_n x^n\tag{1}\label{1},$$ which after plugging-in the specific ...
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### Prove that $\sum_{k} \binom{l}{m+k} \binom{s+k}{n} (-1)^k = (-1)^{l+m} \binom{s-m}{n-l}$ using generating functions

This is an identity from Concrete Mathematics (5.24). Use generating functions to prove that \sum_{k} \binom{l}{m+k} \binom{s+k}{n} (-1)^k = (-1)^{l+m} \binom{s-m}{n-l} \label{...
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### what type of polynomials are these? similar to laguerre polynomial generating function.

I'm studying polynomials generated as coefficients of this generating function: $$[x^n]f(x) = [x^n] \frac{\cos(t\frac{1-x}{1+x})}{(1-x^2)^{\frac{1}{4}}}$$ This is kind of similar to Laguerre ...
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### Countable system of differential equations

I'm interested in solving the following countably infinite system of ODEs \frac{d}{dt}H_{n,k}\left(t\right)=\left(-p\right)nH_{n,k}\left(t\right)+\left(-q\right)nH_{n,k+1}\left(t\...
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### closed form expression for the generating function $\sum_{n=0}^\infty \binom{m+n}{n} x^n$ [closed]
How to find the closed form expression for the generating function: $$\sum_{n=0}^\infty \binom{m+n}{n} x^n\quad \mbox{where}\ m\ \mbox{is a}\ positive\ integer\ ?.$$
We assume that we have a country's currency that contains three coins worth 1, 3, and 4. How many ways can we get an amount of $n$ using these three pieces? In others words what is the number of ...