Articles of generating functions

Series expansion of $\frac{1}{(1+x)(1−x)(1+x^2)(1−x^2)(1+x^3)(1−x^3)\cdots}$?

How would I find the series expansion $\displaystyle\frac{1}{(1+x)(1−x)(1+x^2)(1−x^2)(1+x^3)(1−x^3)\cdots}$ so that it will turn into an infinite power series again??

Transfer Matrix Method to determine the generating function

Let $G = (V,E,\Phi)$ be a weighted directed graph and $\mathcal{W}’ : E \rightarrow \mathbb{C}$ the weighting. Let additionally $m = \# V$, $E_m$ the $m \times m$ identity matrix. Let $v,w \in V$ be in a fixed order in $V$ so $v$ is the i-th and $w$ the j-th element of $V$. Then applies […]

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, but I don’t know how to start from the generating function and get the sequence. Similarly, what if I […]

Kind of basic combinatorical problems and (exponential) generating functions

I have a pretty straightforward combinatorical problem which is an exercise to one paper about generating functions. How many ways are there to get a sum of 14 when 4 distinguishable dice are rolled? So, one die has numbers 1..6 and as dice are distinguishable then we should use exponential generating functions (we count sequences […]

Using Generating Functions to Solve Recursions

I have the recursion $A(n) = A(n-1) + n^2 – n$ with initial conditions $A(0) = 1$. I attempted to solve it using generating functions and I’m not quite sure I have it right, so I thought I might ask if anyone could take a look at my method so far. First I set up […]

Vandermonde-type convolution with geometric term

Is there a closed-form solution to the following sum? \begin{align*} f(r, s, n) = \sum_{k=0}^{n}c^k\binom{r}{k}\binom{s}{n-k} \end{align*} I know this corresponds to find the coefficient of $x^n$ of the generating function \begin{align*} (1+cx)^r(1+x)^s \end{align*} but I don’t know how to proceed from here. Any inputs would be great!

Weak $k$-compositions with each part less than $j$

I am trying to figure out a problem from Richard Stanley’s $\textit{Enumerative Combinatorics}$, which has to do with weak compositions of $n$ (sequence of nonnegative integers whose sum adds up to $n$). The problem is as follows: Let $\kappa(n,j,k)$ be the number of weak compositions of $n$ into $k$ parts, each part less than $j$. […]

What is a generating function?

What is a generating function? In the answer to this question this series comes up. Its generating function is $$A(x) = \sum_{k\ge0} \frac{x^{4^k}}{1-x^{4^k}}$$ Which I took to mean the $x$th element of the series is given by $A(x)$ I guess that is not the case. I tried to read the definition of a generating function […]

Solution to a recurrence relation

Set $F_k(x) := \sum_{n\geq k} S(n,k)x^n$. Prove that $$F_1(x) = \frac{x}{1-x}, \space \space \space F_2(x) = \frac{x^2}{(1-x)(1-2x)} $$ Furthermore, show that the function $F_k(x)$ satisfy the recurrence relation $$F_k(x) = \frac{x}{1-kx}F_{k-1}(x)$$ and solve this recurrence. Any ideas how to approach this problem?

Are there further transformation principles similar to the Inclusion-Exclusion Principle (IEP)?

This question is motivated by the elaboration of the question Combinatorial Proof of Inclusion-Exclusion Principle (IEP). Let’s consider the following two aspects: 1.) IEP transforms at least information to exact information Applying the Inclusion-Exclusion Principle to $n$ sets $A_1,\dots,A_n$ gives: \begin{align*} \left|\bigcup_{j=1}^{n}A_j\right|=\sum_{j=1}^{n}\left|A_j\right| -\sum_{1\leq i \leq j \leq n}\left|A_i\cap A_j\right|\pm\cdots+(-1)^{n-1}\left|\bigcap_{j=1}^{n}A_j\right|\tag{1} \end{align*} We observe that the LHS […]