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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408 views

Generating function solution to previous question $a_{n}=a_{\lfloor n/2\rfloor}+a_{\lfloor n/3 \rfloor}+a_{\lfloor n/6\rfloor}$

In attempting to answer this question, I reduced it to a seemingly simple generating functions question, but after days of work was unable to construct a proof. Since I do not have experience trying ...
7
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168 views

Does $A193201$ count the partitions of $n$ of arbitrary dimension?

By my count, these sequences match for $n=0\ldots6$, where partitions that are the same after relabeling dimensions are considered equivalent (i.e., the dimensions are unordered). For example, for ...
6
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119 views

Specializations of elementary symmetric polynomials

Let $\mathcal{S}_{x}=\{x_{1,},x_{2},\ldots x_{n}\}$ be a set of $n$ indeterminates. The $h^{th}$elementary symmetric polynomial is the sum of all monomials with $h$ factors \begin{eqnarray*} ...
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128 views

Generating Functions, Recursive Polynomials

At the CMFT international conference in Turkey (2009), the following open problem was given: Show that $$p_n(x):=\sum_{k=0}^n \frac{(n-k)^k}{k!}x^{n-k}$$ has only real simple zeros for every $n$. ...
4
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68 views

Help finding a closed form

I have the following function: $$\frac{2e^x}{e^{2x}+1+2x}=\sum_{n=0}^\infty \varepsilon_n\frac{x^n}{n!}$$ I would like to find a closed form for the $\varepsilon_k$. One thing that I do know is that ...
4
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80 views

Generating Function for Associated Stirling Numbers $b(n,k)$

I am trying to identify or find the ordinary generating function (not the exponential generating function) for the Associated Stirling numbers of the Second kind, denoted $$b(1;n,k)=b(n,k)$$ These ...
4
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129 views

Generating functions to solve number of integer solution problem

If I have $x_1 + x_2 + x_3 =10$ with $1\leq x_1 \leq 5, \; 2 \leq x_2 \leq 6, \;3 \leq x_3 \leq 9$ I know that I compute $(t^1+\dots + t^5)(t^2 +\dots + t^6)(t^3+\dots +t^9)$ and look at the ...
4
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125 views

Card game probability

Suppose the following solitaire with a standard deck. I turn four cards visible on the board and on each turn, I remove those suits that appears more than once in the board. Then I fill the board such ...
4
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75 views

The importance of generating series

What follows is a very nebulous question. I just seek for some help in understanding a "technique" which has proven itself very powerful. I noticed that very often generating series appear in ...
4
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89 views

Pairwise sums are equal

The distinct positive integers $a_1,a_2,...,a_n,b_1,b_2,...,b_n$ with $n\ge2$ have the property that the $\binom{n}2$ sums $a_i+a_j$ are the same as the $\binom{n}2$ sums $b_i+b_j$ (in some order). ...
4
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233 views

Generating function for product of binomial coefficients

In general, if $a(n)$ is an integer sequence with generating function $A(t)$ and $b(n)$ is an integer sequence with generating function $B(t)$, it is not easy to find the generating function $C(t)$ ...
4
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207 views

What is the exponential generating function of the inverse matrix of an integer triangle?

Let ${Z}$ denote an integer triangle (like Pascal's), ${Z}_{n,k}$ for $0\leq k \leq n$ and let $f$ be an exponential generating function for polynomials $p_{n}(x)$ with $[x^k]p_{n}(x)= Z_{n,k}$. ...
3
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49 views

What happens to the exponential generating function if the sequence is “stretched”?

Consider a real-valued sequence $(a_0,a_1,a_2,....)$ and the "$n$-stretch"-transformation, which inserts $n$ zeros between each element, for example $n=2$ would represent the transformation ...
3
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44 views

Number of non-negative solutions to an equation - check my work

I tried to solve the following question and would love someone to check my work. Let $\displaystyle x_1,x_2,x_3,x_4$ $100$ digits long numbers where every digit is $1$ or $2$. Find the number ...
3
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117 views

Extinction probability of binomial branching process tends to poisson one.

The folowing is stated and proved in the random graphs book by Luczak, Janson, Rucinski and this is on page 108 in the Giant component section. I can't understand why the conclusion follows from the ...
3
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118 views

Stirling numbers and Power Group Enumeration

The following question is a reference request concerning a derivation of the EGF for the Stirling numbers of the second kind by Power Group Enumeration / Burnside's Lemma, which is $$\sum_{n\ge 0} ...
3
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48 views

Generating series - Finite groups of order $n$

I am wondering if something of interest can be said about one of the two series $$G_1(x)=\sum_{n=1}^{+\infty}{\mathcal{G}(n)z^n}$$ $$G_2(s)=\sum_{n=1}^{+\infty}{\frac{\mathcal{G}(n)}{n^s}}$$ where ...
3
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85 views

Find generating function For sequences

Can anyone out here help? The exercise says: "Find the generating function for each of the sequences below (the general term is given)" Now, the question is how do you find one for those: a) $U_n = ...
3
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124 views

Generating Functions - Number of numbers with sum of digits $\leq 7$

How many natural numbers with $n$ digits are there, where the sum of their digits is $\leq 7$? So I said the following: For the first digit (MSB) we can't have a zero, and all the other $n-1$ ...
3
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161 views

Combinatorics: Distributions with several constraints - potential generating sequences?

This seems to be a "stars and bars problem" - however I am unsure how to apply a constraint. Furthermore, I would like to understand how this would be done in both counting as well as generating ...
3
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557 views

Asymptotics for the expected length of the longest streak of heads.

As Introduction to Algorithms (CLRS) describes, the problem is Suppose you flip a fair coin $n$ times. What is the longest streak of consecutive heads that you expect to see? The book claims ...
2
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24 views

Asymptotic analysis of coefficients of ordinary generating functions with radius of convergence $1$ seems to always predict polynomial growth rate

Wikipedia gives the following formula for obtaining asymptotic information about the coefficients of an ordinary generating function from information about the generating function itself: if the ...
2
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30 views

Multivariable generating functions

Let's consider a 2-variable generating function for the Dyck triangle numbers. Reccurence, satisfying to the triangle conditions is $d_{n, k}=d_{n-1, k-1}+d_{n-1, k}+d_{n-1, k-1}$, $d_{0, 0}=1$, ...
2
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28 views

Formal sum of product of all size k subsets of a set

Is there a nice way to use generating functions to represent the formal sum of all size-$k$ subsets of a set $S$? Here I want to represent a subset by the product of its elements. For example, if $S ...
2
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22 views

Link between partition function and ordered partition function

The partition function $p(n)$ measures the number of partitions of $n$, or the number of ways in which natural numbers can be summed to produce $n$, without regard to order. For example, the ...
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34 views

Don't understand why this generating function needs to be taken to the power $-1$

I'm given the following recursive formula for a sequence: $u_{n+1}=u_n+\sum_{k=1}^{n-1}u_ku_{n-k}\ (n\ge1),\ u_0=0,\ u_1=1$ Since $u_0=0$ we can rewrite: $u_{n+1}=u_n+\sum_{k=0}^{n}u_ku_{n-k},\ ...
2
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52 views

Is there a systematic relation between the generating functions for the rows vs that for columns of infinite sized Carleman-matrices?

(Roughly related, but generalizing, of this earlier question) Background: The first part of the following(the column-wise-focus) is also described in Eri Jabotinski's 1953-treatize Representation of ...
2
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34 views

Transforming Exponential to Ordinary Generating Functions

I am looking for a particular ordinary generating function, if it exists for the Associated Stirling Numbers of the second kind $$b(1;n,j)=b(n,j)=\sum_{k=0}^j(-1)^k\binom{n}{k}S(n-k,j-k)$$ Where ...
2
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53 views

How to solve this tough recurrence relation?

For solving a related probability problem, I'm required to solve the following recurrence relation:- $q(r,b,L)=\frac{b}{r+b}q(r,b-1,L)+\sum_{k=1}^{L}\frac{r}{r+b}\times\cdots \times ...
2
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60 views

Correctness of counting with product of generating functions

In generating functions we can related the coefficients of the generating function with as sequence. $$f(x) = \sum^{\infty}_{n=0}f_nx^n$$ and the sequence that corresponds to it: $$( \ f_0 \ , \ ...
2
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73 views

Partitions of 200 into at most 6 parts.

I'm working on a partition problem, and I got an answer that is simply staggering, and I was hoping for someone to verify whether my answer is correct. The question was simply to determine the number ...
2
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53 views

Do asymptotically equivalent coefficients survive convolution at least in Θ?

This is a follow-up question to this one where I asked if asymptotic equivalence of coefficients carried over after convolution, resp. why this was not the case. Answerer Daniel Fischer proposed that ...
2
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83 views

Puzzle with character order

Suppose I have 3 letters a, b, c and I want to find the minimum length of a string that uses all the double combinations of the aforementioned letters. How should I do it or how are such problems ...
2
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55 views

Deriving combinatorial coefficients for a series related to the binomial series (but with a third index)

I have a problem that reminds me a lot of the binomial coefficients, and I'm wondering how to compute future elements with a nice formula, in the same way that $\binom{n}{k}$ is used as a symbol ...
2
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67 views

Find the sum of exponentails of squares $\sum_{r=1}^n e^{-\alpha r^2}$

I would like to find $$a_n =\sum_{r=1}^n e^{-\alpha r^2},\qquad \alpha\in\mathbb{R}$$ I tried to solve the equivalent recursion $$a_n=a_{n-1}+e^{-\alpha n^2}\quad(n>0),\qquad a_0=0.$$ with an ...
2
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68 views

Question about generating functions with trig

Consider there recurrence relation $$a_n=\frac{5}{4} a_{n-1} \cos{(\pi\cdot a_{n-1})}.$$ I am trying to find the generating function $A(x)=\sum_{n=1}^\infty a_nx^n.$ I've tried the following: ...
2
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65 views

Distributing problem using generating functions

For $r\in\mathbb{Z^*}$, let $a_r$ denote the number of ways to distribute $r$ identical objects into $3$ identical boxes, $b_r$ be the number of distributions so that the boxes are to be non-empty, ...
2
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47 views

Sequence of Ratios

Let $\{a_n\}_{n\ge 0}$ be a positive real sequence and define $$r_n=\frac{a_{n+1}}{a_n},\quad n\ge 0$$ Suppose that we know the formal power series of $a_n$, i.e. we know the following: ...
2
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102 views

Given a generating function for $\sum a_n z^n$, what is the generating function of $\sum a_n^2 z^n$

Given a generating function $G(z)$ for $\sum_{n=0}^{\infty} a_n z^n$, what could be said about the generating function of $\sum_{n=0}^{\infty} a_n^2 z^n$, what algebraic form should it have? For ...
2
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46 views

Generating Function for rational sequence

I'm trying to compute the generating function for the function defined for $1\le N < \beta$, $C_N = \frac{\beta}{N(\beta-N)}$. I think my math so far is correct, but I don't know how to solve the ...
2
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287 views

Prove that sum is finite with the help of generating function

Please help me to prove that the following sum is finite $$ \sum_{j=2l-2}^{\infty}j!\, a_j^{(l)}, $$ here the generating function of $\displaystyle{a_j^{(l)}}$ is ...
2
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203 views

What is the closed form of generating function of a power law?

I want to know if there is a "closed form" of the following generating function, $G_n(x) = \sum_{n=0}^{\infty} P_n x^n$ where, $P_n = C(n_0 + n)^{-\gamma}$ where $C$ is a normalization constant, ...
2
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132 views

Help on Generating function - Analysis of Median of three - Quick Select

I am trying to find out the average case analysis of median of 3 - quick select. The recurrence relation is \begin{equation} C_{n,j} = 1 + \sum_{k=1}^{j-1}\pi_{n,k}C_{n-k,j-k} + \sum_{k=j+1}^{n} ...
2
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91 views

Proving generating functions equality

What do you use to prove the following equality (and possibly more general ones of the kind)? \begin{align*}\sum_{r,s,t} \frac{q^{r^2+rs+s^2+st+t^2}}{(q)_r (q)_s (q)_t} z_1^{r+s} z_2^{s+t} = ...
2
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234 views

Generating Function of Integer Partition Such that at Least One Part is Even

I've been having a few issues coming up with a generating function for an integer partition such that at least one part is even. What I have got so far is: The generating function with no ...
2
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56 views

Which type of counting problems are solved by hypergeometric function as generating function?

Which type of counting problems are solved by hypergeometric function as generating function? would you mind giving some examples such as relating counting with hypergeometric function as generating ...
2
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144 views

Asymptotics of shifted Catalan numbers

I am trying to understand a lemma from a paper (most of the proof was omitted), and I've got it melted down to the following: Let $B_n$ denote the $n$-th Catalan Number. In a sufficiently small ...
2
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296 views

Deep understanding on exponential generating function

In spite of having done some exercises, I still find it harder to understand exponential generating function deeply than ordinary generating function. Could someone explain it "deeply"? Or are there ...
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31 views

Generating function of a_n * b_n

I've been searching for the answer yet no luck. Please let me know if this is a duplicate. Let $A(x) = \sum a_n x^n$, $B(x) = \sum b_n x^n$, $C(x) = \sum a_n b_n x^n$. I am looking for the closed ...
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Elusive closed form for card permutation problem

Does a closed form formula f(n) exist for the two rightmost columns? The two question marks are meant to be 0. The diagram is a summary of the numerical results from original question: Permutations ...