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.
4,699 questions
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Find the number of quadruples of non negative integers $(a,b,c,d)$ such that $105a + 70b + 42c + 30d = 2025$
Find the number of quadruples of non negative integers (a,b,c,d) such that $105a + 70b + 42c + 30d = 2025$
Simply using generating functions makes things very lengthy:
This problem is similar to ...
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How do I solve this recurrence relation with a squared coefficient?
I started with the following recurrence relation
$$ s \left( n \right) = s \left( n-1 \right)^{2} - 1, \qquad s \left( 0 \right) = 2$$
and got to
$$\sum_{n=0}^{\infty}s\left(n+1\right)x^{n}=\sum_{n=0}^...
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bijective proof of identity coefficient-extracted from negative-exponent Vandermonde identity, and the upper-triangular Stirling transforms
Context: Mircea Dan Rus's 2025 paper Yet another note on notation (a spiritual sequel to Knuth's 1991 paper Two notes on notation) introduces the syntax $x^{\{n\}}=x!{n\brace x}$ to denote the number ...
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Extracting coefficients from generating function that is not well-defined in the zero point
In a physics problem I'm currently considering generating functions containing the term $1/\sqrt{t^2}$, where earlier in the derivation I have restricted my attention to the cases $\lvert xt\rvert<\...
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Is it possible to solve recurrence relation with non-constant coefficient when coefficient is not explicitly given in terms of $n$?
I have a recurrence relationship as:
$$F_{n}=a_nF_{n+1}+a_nF_{n-1}$$
Is it possible to solve such a relation (using a generating function) when the explicit value of $a_n$ is given but not explicitly ...
- 651
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When does this series converge to a positive integer
I have been working on different unsolved problems of mathematics, mainly for fun and to get a feel for what makes them so difficult. In the process, I stumbled upon this problem, and I am stuck:
Let $...
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Conjectured recurrence for $n$-th row of Salie numbers independently of other rows
Let
$B_n$ be the $n$-th Bernoulli number.
$T(n,k)$ be A065547, i.e., an integer coefficients known as triangle of Salie numbers whose exponential generating function satisfies $$ \sum\limits_{n=0}^{\...
user929945
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Exponential generating function for even Eulerian polynomials
Background
The Eulerian polynomials $A_{n}(\cdot) $ have the following exponential generating function (e.g.f.):
$$ \sum_{n=0}^{\infty} A_{n}(t) \frac{x^{n}}{n!} = \frac{t-1}{t-e^{(t-1)x}} \ . \tag{1}\...
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Asymptotic approximation of an integer sequence
MOTIVATION
I think that computing the asymptotics of $F(x)$ in this answer would give a solution for this question.
QUESTION
For $n \ge 0$, let $u_{2n} = \binom{2n}{n}^2 / 4^{2n}$ and $u_{2n+1} = 0$. ...
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New recurrence relation with unusual pattern Is there a closed-form solution? [closed]
I came across an interesting recurrence relation that I couldn’t find any references for:
$$
a_1 = 1, \quad a_{n} = \frac{1}{n^2} \sum_{k=1}^{n-1} k \, a_k \, a_{n-k} \quad \text{for } n \geq 2
$$
It ...
- 177
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Symmetrization of functional derivatives
I'm dealing with a mathematical problem stemming from quantum field theory (QFT). However, at the moment, I'm not concerned with the physics aspect of it and, hence, I wish to view it in purely formal ...
user724377
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Integer coefficients defined recursively with main diagonal equals A195979
Let
$A_n(x)$ be the family of exponential generating fucntions such that $$ A'_n(x) = nA_n(x) + A_{n-1}(x), \\ A_n(0) = 1, A_0(x) = 1. $$
$T(n,k)$ be an integer coefficients whose exponential ...
user929945
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A combinatorial identity containing double sums: where to find it? [closed]
When I try to establish an explicit formula for computing the generalized Euler polynomials, I encounter with the following identity
\begin{equation*}
\sum_{m=0}^j\frac{(-1)^m}{2^m}\binom{j}{m}\sum_{...
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Applying generating function coefficient to find total arrangements
So we have $12$ or fewer balls and $3$ boxes and he has to fill the boxes with the balls with the following constraints -
No box is empty
Box $B$ has at least $3$ balls &
Box $C$ has at most $5$ ...
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counting number of integer solutions
Given $m,h$ we need to count the nonnegative solutions of $n_0 + n_1 + n_2 + n_3 + \dots + n_h = h$ subject to the constraint $\sum_{i = 0} in_i \equiv 0$ mod $m$. I have tried to use generating ...
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Expansion identity for the Eulerian polynomials of the second order
Background
$\newcommand{\polylog}{\mathrm{PolyLog}}$
The Eulerian polynomials $A_{m}(\cdot)$ are defined by the exponential generating function:
\begin{equation}
\frac{1-x}{1-x \exp[ t(1-x) ] } = \...
- 7,768
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How to prove the identity between two generating functions involving binomial coefficients?
Let
$$ \left(\sum_{k\geq 0} \binom{3k} k x^k \right)^4 = \sum_{k\geq 0} a_k x^k $$
and
$$ \log\left( \sum_{k\geq 0} \binom{3k+3} k x^k \right) = \sum_{k\geq 1} b_k \frac{x^k}{k}. $$
For example, the ...
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In how many ways can one pay 2$ with coins of 5,10 and 50 cents? (not american currency, there are 5 ,10 and 50 cents coins)
I had come as far as creating the first steps of building a generating function for the number of ways to pay n dollars with the coins,
$x_1$- num of 5 cent coins
$x_2$- num of 10 cent coins
$x_3$- ...
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Finding $\lim_{n \to \infty} \sum_{k=n}^{5n} \binom{k-1}{n-1}(1-x)^nx^{\,k-n}$
Recently I came across a problem that has me stumped. It is as follows:
Find the limit
$$
\lim_{n \to \infty} \sum_{k=n}^{5n} \binom{k-1}{n-1}(1-x)^nx^{\,k-n},\,\, x \in [0,1].
$$
I spent a long ...
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Expected number of black tiles in $3 \times n$ grid when no $3$ in line
Background: this question which deals with the maximum number of black tiles.
Let's consider all the ways to paint the tiles of an $3 \times n$ grid white/black so that no three black tiles are in a ...
- 12.6k
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$\mathcal{P}$-rook polynomial of a grid [duplicate]
The $\mathcal{P}$-rook polynomial of a polyomino $\mathcal{P}$ is
$$
r_\mathcal{P}(T) = \sum_{k=0}^{r(\mathcal{P})} r_k(\mathcal{P})\ T^k,
$$ where $r_k(\mathcal{P})$ is the number of ways to place $...
user969954
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Why generating function of Catalan numbers only defined on $\vert x\vert<\frac{1}{4}$?
I am reading about Catalan number and their generating function but I donot understand the following.
Why is
$$C(x)=\sum_{n=0}^{}C_{n}x^{n}=\frac{1-\sqrt{1-4x}}{2x}$$
only defined for $\vert x\vert \...
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Is there a closed form for the generating function for partitions of $n$ into up to $m$ unequal parts?
Wondering if there is a closed form for the generating function for partitions of $n$ into up to $m$ distinct parts.
For up to $m$ non-distinct parts, the generating function is just the simple
$[x^n]$...
- 590
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Power series expansion involving exponential and binomial coefficients
It arose from an engineering context concerning power intake structures at the Mica Dam in British Columbia.
The engineers encountered a function of the form:
$$
f(u) = \frac{\exp(u + nu) + \exp(-nu)}{...
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Asymptotics of $\displaystyle\sum_{i=1}^k i^{-2} (1-i^{-2})^n$ as $k\to +\infty$
For fixed $n$ a non-negative integer, I tried deriving the complete asymptotic expansion of,
$$S_k :=\sum_{i=1}^k i^{-2} (1-i^{-2})^n$$
as $k\to+\infty$ using generating functions and complex analysis,...
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Find the number of positive integer solutions to the equation $x_1+x_2+x_3+x_4+x_5=2024$ such that $x_2 > 5$ and $x_1$ is divisible by 3
Problem. Find the number of positive integer solutions to the equation $x_1+x_2+x_3+x_4+x_5=2024$ such that $x_2>5$ and $x_1$ is divisible by 3.
I am going to use generating functions to solve this ...
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How to derive Krattenthaler's explicit formula for the coefficients of Motzkin/Dyck path generating function
Equations (10.73) and (10.74) in Krattenthaler's Lattice Path Enumeration (from Bona's Handbook of Enumerative Combinatorics), give the generating functions for the counts of Dyck paths and Motzkin ...
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Asymptotics of $a_n$ such that $n^2(a_{n+1}-2a_n+a_{n-1})=\lambda a_n$
Given $\lambda, a_0, a_1\in\mathbb{C}$, define $a_n$ for $n>1$ by the recurrence
$$
n^2(a_{n+1}-2a_n+a_{n-1})=\lambda a_n.\qquad(n>0)
$$
What is (the main term of) the asymptotics of $a_n$ as $n\...
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Generating Functions and Recurrence Relations
I am currently reading Paul Zeitz's "The Art and Craft of Problem Solving, Third Edition" and I was confused on one of the examples.
"Define the sequence $a_{n}$ by $a_0 = 1$ and $a_n = ...
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prove that $S\bigg(\frac{a}{1-a},\frac{b}{1-b},\frac{c}{1-c}\bigg)=\frac{1}{1-(a+b+c)}$
Assume that $S(a,b,c)$ represents the generating functions of all words consisting of alphabet ${a,b,c}$ such that no consecutive same letter allowed.
Then,for illustration, $$S(a,b,c)=\epsilon +a+b+c+...
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A dice is rolled 31 times in a row. How many times does the sum of the results amount to 38? [closed]
I reached as far as finding the following generating function, but still I am unable to solve this question.
$$
\left[ x^{38} \right] \left( \frac{x(1 - x^6)}{1 - x} \right)^{31}
$$
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The irrationality/transcendence of $\prod_{n=0}^\infty(1+1/n!)$
Is the following number demonstrably irrational? Is it demonstrably transcendental?
$$
\prod_{n=0}^{\infty} \left(1 + \frac{1}{n!} \right)
= 7.36430827236725725637277250963105\dots
= 7 + \frac{1}{2 + \...
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Generating Functions - Number of numbers with sum of digits =18
The problem is : how many numbers with n digits at most, are with the digit sum = 18?
For the first digit we can't have zero and all the other $n−1$
digits they can be anything from $[0,..9]$
so, $$F(...
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Why introduce $F[[x]]$ for infinite generating functions?
My instructor assigned us a topic on generating functions, and I’m currently reading some materials to try to understand them.
For finite sequences, I have observed that the generating function method ...
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Reference for Chebyshev Polynomial (of the second kind) Identity
I am looking for a reference for the following identity involving Chebyshev polynomials of the second kind:
$$ \frac{\left(U_{n-1}\left(\frac{1-x}{2x}\right)\right)^2}{xU_{2n-1}\left(\frac{1-x}{2x}\...
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From Algebra to a Lie Algebra element, a concrete combinatorics problem
I am working on a physics problem and I was able to reduce it to a "simple" combinatorial formula, the following one:
$$ \mathcal{I}_n = \sum_{\sigma\in S_n} \frac{(-1)^{s_\sigma+1}}{\binom{...
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Doubt regarding generating functions of surjections with restrictions
The following question is from Bóna's "A Walk Through Combinatorics".
Let $s_0 = 1$, and for n > 0, let $s_n$ be the numbers of all surjections from [n] to [k], where n is a fixed ...
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Asymptotic Behavior of a Recurrence Sequence
During my studies, I stumbled upon the following recurrence.
\begin{align*}
a_n = 2 \sum_{k=0}^n \binom{n}{k} \frac{a_k}{2^n}, \qquad n \ge 2,
\end{align*}
with initial conditions $a_0=a_1=1$.
I ...
- 1,157
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Equality of two multivariable generating functions if their closed formulas for a single variable are same
Disclaimer: I am physics student interesting in mathematics, especially combinatorics. I dont have deep mathematical background, so I registered this community to learn. I wrote this disclaimer ...
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Weighted Polynomial Inequality
Let $n > m \geq 1$ be integers, and let $x, y \geq 0$.
Consider the following inequality:
$$
m \left( \sum_{i = 0}^{m} \frac{1}{i + 1} \binom{m}{i} x^{m - i} y^i \right)
\left( \sum_{i = 0}^{n} \...
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Closed form of an infinite sum over a closed form of a generating function?
I would like to know if a sum of the general type
$$\sum_{t=0}^{\infty} \frac{x^{at^2+bt+c}}{1-x^{dt+e}} $$
can be solved ($a,b,c,d,e$ being integers)? Maybe only for summing up finitely many terms ...
- 127
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Can someone explain the details of the derivation of Binet’s formula (Fibonacci numbers) with Z-Transform?
I have serious issues with the derivation that I see everywhere. I’ve searched everywhere but couldn’t find a satisfactory answer. My questions are at the bottom but I’ll first show what the general ...
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Showing $\sum_{k=0}^n\frac{(-1)^k}{k+1}\binom{n}{k}^2=\frac{(-1)^{n/2}\,n!}{(n+1) ((n/2)!)^2} $ for even $n$
I am stuck on solving the following:
If $n$ is an even integer, show that $$\binom{n}{0}^2 - \frac{1}{2} \binom{n}{1}^2 + \frac{1}{3} \binom{n}{2}^2 - \cdots + \frac{(-1)^n}{n+1} \binom{n}{n}^2 = \...
- 879
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The meaning of $\exp\left(\frac{d}{dt} \pm \frac{d^2}{dt^2} \right)$
I'm solving a problem from a book "Introduction to discrete mathematics" by S. K. Lando. The problem is to find the action of $$\exp\left(\frac{d}{dt}\pm \frac{d^2}{dt^2}\right)$$
on ...
- 1,486
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Coloring $1\times n$ board with $3$ colors, each color appearing an even and nonzero number of times
I am asked to solve the following problem:
Let $h_n$ be the number of ways to color the squares of a $1\times n$ board with red, white, and blue such that each color appears a nonzero, even number of ...
- 121
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Special coefficient of $q$-binomial
I'm trying to determine the coefficient of $q^{k^2-k}$ in the $q$-binomial
$$\binom{2k}{k}_q.$$
So far, I've simplified it to
$$\binom{2k}{k}_q = \frac{[2k]_q!}{([k]_q!)^2} = \frac{(1-q^{2k})\cdots(1-...
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A direct proof of $\ln((1+s)(1+t))=\ln(1+s)+\ln(1+t)$ in formal power series
I want to prove that
$$\ln((1+s)(1+t))=\ln(1+s)+\ln(1+t)$$
where $\ln(1+t)=t-\frac{t^2}{2}+\frac{t^3}{3}+...$ is the formal power series of logarithm. I know that
$$\frac{d}{dt}\ln(1+t)=\frac{1}{1+t}$$...
- 1,486
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0
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Three-variable recurrence relation with rotated indices
I'm looking for help in solving the recurrence relation
$$ a_{i,j,k}=1+p\,a_{j+1,k,i-1}+q\,a_{j,k+1,i-1}\quad\quad\quad\text{ for }i\ge1,\;j\ge1,\;k\ge0$$
with boundary conditions
\begin{align*}
...
- 2,357
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0
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Reducing rationality of Igusa's zeta function for system of equations to a single equation.
Let $F$ be a $p$-adic field and $f_1,\dots,f_r$ be non-constant polynomials over $F$ with integral coefficients. Let $N_n$ be the number of $\mod {\mathfrak p}^n$ solutions to $f_1,\dots,f_r\equiv0\...
- 8,908
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1
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Partitioning the set of non-negative integers in a certain way.
Context: I have been self-learning the theory of generating functions for my combinatorics class. And I have been solving these problems from MIT OCW. I have done all problems till $10$ on my own but ...
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