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3
votes
0answers
97 views

Involutions on $[0,1]$ given by power series (related to probability generating functions)

Let $A$ be a function from $[0,1]$ to $[0,1]$. $A$ is an involution if $A(A(x))=x$ for all $x\in[0,1]$. Which involutions $A$ exist such that $A(x)=\sum_{k=0}^\infty a_k x^k$ with $a_0=1$ and ...
0
votes
1answer
106 views

The number of ways of writing a natural number as the sum of distinct elements from a finite set [closed]

Given a finite set of natural numbers $A$, is there a generating function for the number of ways of writing a given natural number $n$ as a sum of $s$ distinct elements from $A$?
1
vote
1answer
52 views

Restricted partitions with square terms only

Let $P(N,M,n)$ be the number of partitions of $n$ such that each term is $\le N$ and there are at most $M$ terms. So we know the generating function for $P(N,M,n)$ is $ \frac{(q)_{N+M}}{(q)_M ...
2
votes
1answer
65 views

Generating function of alternating even terms in the Vandermonde Convolution

I have \begin{equation} G(x) = \sum_{i = 0}^{\infty} \sum_{r = 0}^{\infty} (-1)^i \binom{k}{2i} \binom{n-k}{r} x^{2i} x^{r} = \frac{1}{2} \left( (1 + x{\iota})^{k} + (1 - x \iota)^{k} \right)(1 + ...
2
votes
1answer
77 views

Any interesting properties of the matrix $M:=(m_{ij})$ with $m_{ij}=min(i,j)$? [closed]

Do you know any interesting properties on the matrix $M(n):=(m_{ij})$ of size $n \times n,$ where $m_{ij}= \text{min}(i,j)$ ? The matrix $M$ enumerates certain combinatorial objects.
2
votes
1answer
346 views

A particularly “natural” algebraic structure with three commutative, pairwise-distributive operations

EDIT: As mentioned in my answer below, I was mistaken in thinking Dirichlet convolution distributes over ordinary convolution. I'm leaving this question here for reference. I keep stumbling on the ...
5
votes
1answer
96 views

Asymptotics of a Bivariate Generating Function

I have the following generating function, $$G(x,y)=\sum_{n,k \geq 0}a(n,k)x^ny^k = \frac{(y^2-y)x+1}{(y-y^3)x^2-(y+1)x+1}$$ and I am interested in obtaining an asymptotic for the sequence $a(n,k)$ ...
1
vote
2answers
138 views

How do powers affect asymptotics in generating functions?

Let $a_n$ be a sequence of non-negative real numbers, and $A(x) = \sum_{n=0}^{\infty} a_n \frac{x^n}{n!}$ its exponential generating function. Also, suppose $B(x) = \sum_{n=0}^{\infty} b_n ...
8
votes
0answers
87 views

types of generating functions (ordinary, exponential, ???) closed under substitution

A nice feature of ordinary and exponential generating functions is that they are closed under substitution: if $F(z)$ and $G(z)$ both have integer coefficients, then $F(G(z))$ also has integer ...
0
votes
1answer
43 views

Solving the difference equation $h(\vec x)\cdot A(\vec x)=\sum_{i=1}^m A(\vec x - \vec e_i)$

I am trying to solve the following difference equation: $$h(x_1,x_2,x_3,x_4) \cdot A(x_1,x_2,x_3,x_4)=A(x_1-1,x_2,x_3,x_4)+A(x_1,x_2-1,x_3,x_4)+A(x_1,x_2,x_3-1,x_4)+A(x_1,x_2,x_3,x_4-1)$$ with ...
12
votes
2answers
305 views

Asymptotics of coefficients of implicitely defined generating function

I have two integer sequences $\{a_n\}_{n=0}^\infty$ and $\{b_n\}_{n=0}^\infty$. Explicit formulas for the $a_n$ are known and their asymptotic growth is fully understood. My wish is to also understand ...
3
votes
0answers
42 views

Using Mellin transform for a certain function

In short, I want to use the Mellin transform to obtain the asymptotic behavior of the sequence $D_n = \frac{ [z^n] D(z)} {C_n}$ where $$ D(z) = \frac 1{2z}\sum_{p \ge 1}C_p \left( ...
4
votes
1answer
157 views

Raising coefficients of a power series to some power

Suppose you are given a power series $P=\sum_{i=0}^\infty{a_nt^n}$. I am primarily concerned with those power series coming from rational functions of the form $$ ...
11
votes
1answer
363 views

Number of trivializations of a trivial word in the free group

Let $M$ be the free monoid on $2n$ generators $x_1,X_1,...,x_n,X_n$ and consider the set $T$ of all those elements of $M$ which map to 1 of the free group on $x_1,...,x_n$ under the homomorphism $\pi$ ...
1
vote
1answer
58 views

Generating function for products of laguerre polynomials?

In a quantum physics context, I would like to evaluate $S=\sum_{n=0}^\infty z^n\cos(\pi L_n(x))$ for $z<1$. I found generating functions for squares of Laguerre polynomials but not for any higher ...
2
votes
2answers
165 views

How to calculate one Cauchy type determinant

As we know, a Cauchy determinant of size n admits the following explicit formula: $$\det \left(\frac{1}{x _i+y _j}\right) _{1\le i,j \le n}=\frac{\prod _{1\le i < j\le n} (x _j-x _i)(y _j-y ...
0
votes
0answers
143 views

Generating function for reciprocals of Stirling numbers?

Is there an (ordinary or exponential) generating function for the reciprocals $\frac{1}{s(n,k)}$ of the Stirling numbers of the first kind? Also, is there some general way to find generating ...
3
votes
0answers
233 views

Solving a doubly exponential generating function

I am analyzing the average time complexity of some algorithm on some probabilistic model, and I've come to a doubly exponential sequence for which I cannot find corresponding generating function. I ...
1
vote
1answer
141 views

Proof of closed walk generating function identity

In 'Spectral Conditions for the Reconstructibility of a Graph' Godsil and McKay give a short proof of an identity (Lemma 2.1) that relates the generating function for the number of closed walks ...
8
votes
1answer
206 views

Higher moments of information and Renyi entropy

For a given discrete probability distribution, Shannon entropy can be though as an expectation value $\langle - \log p \rangle$ (see also: What is entropy, really?, What is the role of the logarithm ...
0
votes
0answers
102 views

Help solving a recurrence relation

For solving a related probability problem, I need 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 ...
1
vote
1answer
193 views

Name for series $\sum f_n x^n / (n! (n+k)!)$

Let $(f_n)_{n\ge0}$ be a real sequence. Then $\sum f_n {x^n \over n!}$ is called the exponential generating function of $(f_n)$. Let $k\ge0$ be a nonnegative integer. If we add another factorial ...
4
votes
0answers
93 views

Prove a complicated function (in epidemic spreading search) to be convex

When analyse epidemic spreading, I came across to prove that a complicated function $f(x)$ is convex when $0 \leq x \leq 1$. \begin{equation} f(x)=\frac{b_1g'(x) f_1(x)^{n-2}+g'(1) \gamma}{g'(1) ( ...
0
votes
0answers
50 views

Coupled recurrence relations for generating functions, involving squared arguments

I'm trying to find a solution for the following system of three equations in terms of three bivariate generating functions. $G(x,y)=b(x,y) \cdot \left( G(x,y^2) + I(x,y^2) \right) +y ;$ $H(x,y)=x ...
4
votes
2answers
333 views

An interesting calculation of derivative

I was trying to get the probability distribution $p(n)$ from a generating function $G(s)$ like this: $G(s) = e^{a(s-1)^2}=\sum s^np(n)$ I need first to do Maclaurin expansion of the exponential and ...
7
votes
2answers
347 views

What is the number of noncrossing acyclic digraphs?

A noncrossing graph on $n$ vertices is a graph drawn on $n$ points numbered from $1$ to $n$ in counter-clockwise order on a circle such that the edges lie entirely within the circle and do not cross ...
1
vote
0answers
138 views

Solving a recurrence (with the form of a convolution) involving binomial coefficients

While dealing with a problem related to intersection of hyperplanes I have come across the following recurrence to obtain the values of $K_{j}$ \begin{array}{cccccccccc} 1 & = & ...
9
votes
4answers
1k views

Ordinary Generating Function for Bell Numbers

In the OEIS entry for Bell numbers, there appears a generating function $$\sum_{k=0}^\infty B_k t^k = \sum_{r=0}^\infty \prod_{i=1}^r \frac{t}{1-it}$$ However, I could not locate any proof of ...
3
votes
1answer
260 views

Is there a nice way to write the generating function obtained by taking the quadratic coefficients of another one?

Suppose that you have a generating function $$ f(q) = \sum_{k=0}^\infty a_k q^k $$ It's not too hard to obtain the generating function $$ f_{n,m}(q) = \sum_{k=0}^\infty a_{nk + m}q^k $$ by taking a ...
5
votes
1answer
259 views

Examples of 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*} ...
1
vote
1answer
167 views

Matrix Generator for M/M/1 Queue Waiting Time Distribution

I "believe" that generator, $\bf W$, of the waiting time distribution for the M/M/1 queue is given by the following (I'm not sure if this is even correct): ${\bf W} =\left( \begin{array}{ccccc} 0 ...
1
vote
1answer
68 views

Approaches to implicitly defining generating function

First,every language in Chomsky hierarchy(or c.e.language) corresponds to a generating function,the set of the functions is GF,now,a question : is every generating function with integral coefficient ...
0
votes
0answers
102 views

Estimating when does a certain binomial sum exceed an upper bound

Given a fixed integer $n > 0$ and $0 \le m \le n$ let us define the numbers $$f_{n,m} = \sum_{i=\lfloor m/2 \rfloor}^m {n-2i \choose n - m -i}{i+1 \choose m - i +1}.$$ For example $f_{n,0} = ...
10
votes
1answer
273 views

Laurent polynomials associated to partitions and a $Q$-deformation of $\sigma(d)$

Let $\alpha \vdash d$ be a partition of $d$, i.e. $\alpha = (\alpha_1 \geq \alpha_2 \geq …\geq \alpha_l)$, where $\sum_k \alpha_k = d$. Define a Laurent polynomial in $Q$ as follows: $$ P_\alpha(Q) = ...
1
vote
0answers
136 views

How to prove this identity? (Perhaps related to partition) [closed]

How to prove this identity? $ \sum_{n\ge 0} \frac{x^{n^2}}{(1-x)(1-x^2)\cdots(1-x^n)}= \frac{1}{\prod_{k \ge 0}(1-x^{5k+1})(1-x^{5k+4})}$ I will appreciate it a lot if a solution using method ...
4
votes
0answers
140 views

Asymptotics and combinatorics

Wright's expansion of $$ (1-z)^b\exp[A/(1-z)^c], \text{for } A > 0, 0<c<1\tag{1} $$ is, in the words of the late, great Mark Kac "well known to those that know it well". (See, for example, ...
1
vote
0answers
119 views

Is there inverse FFT algorithm for Fourier transform of a integer-valued random variable?

In many applications, it is possible to derive an explicit expression for the Fourier transform of a random variable $X$ $$\varphi (\theta ) = \sum\limits_{n = 0}^\infty {{p_n}} {e^{in\theta }}$$ ...
2
votes
1answer
596 views

Probability generating function zero implies random variable is infinite

Let $V$ be a random variable supported on the nonnegative integers (including $\infty$) and $f(x) = \mathbf E x^V$ be the probability generating function. In our model $V$ is the number of visits to ...
0
votes
3answers
152 views

Hypergeometric sum specific value

How to show? $${}_2F_1(1,1;\frac{1}{2}, \frac{1}{2}) = 2 + \frac{\pi}{2} $$ It numerically is very close, came up when evaluating: $$ \frac{1}{1} + \frac{1 \times 2}{1 \times 3} + \frac{1 \times 2 ...
2
votes
1answer
265 views

An infinite product: combinatorial interpretation

It is an undergraduate exercise to show that the generating function for the sequence of unrestricted integer partitions $p(n)$ is the celebrated infinite product ...
2
votes
2answers
1k views

Number of 1 in binary representation of n

Let $1(n)$ be the number of digits $1$ in binary representation of number $n$. For example, $13=1101_2$ so $1(13)=3\\$ Is there explicit form of $\,\,\sum{1(i)x^i} $? I checked OEIS and didn't find ...
5
votes
1answer
175 views

The number of partitions between two fixed partitions

Given two partitions M and N, with $M_i \leq N_i$ for all $1\leq i\leq \max\{l(M),l(N)\}$. Is there a formula for the generating function: $$\sum_{\lambda: M_i\leq \lambda_i\leq N_i} ...
-1
votes
1answer
258 views

Does anyone recognize this generating function [closed]

$a_1=1, a_2=1, a_3=3, a_4=15, a_5=105$ Reccurence formula is $a_{k+1}=\sum\limits_{\lambda_1+\lambda_2+\ldots+\lambda_s=k,\ \lambda_i\geq1} a_{\lambda_1}a_{\lambda_2}...a_{\lambda_s}{k \choose ...
0
votes
0answers
104 views

What are the properties of this linear operator?

Suppose $f(x)$ is a function which satisfies the following condition: $$f(x)=\sum_{k=0}^\infty G(2k)\frac{x^{2k}}{(2k)!}$$ Where the generating function $G(x)$ is a "natural" or "discrete-analytic" ...
5
votes
2answers
180 views

Quantities whose generating functions are symmetric

This is inspired by an old Putnam problem from 2005, and a solution given by Professor Greg Martin (a Professor of Mathematics at the University of British Columbia, also a user on MO). The question ...
2
votes
0answers
83 views

Lagrangean equations for the generating function of quadrangulations

Let $M(z)$ be the generating function of edge-rooted connected quadrangulations, with $z$ marking the number of edges. I derived the following Lagrangean equations for $M(z)$: $$M(z) = ...
1
vote
1answer
131 views

Truncated sums of symmetric polynomials; reference request for an algebraic derivation

EDIT: This is a case of being too wrapped up in a formulation ($e_j,p_i,$ and the like) to try something simple. It did not occur to me to pull exp to the outside in the weeks I have stared at this. ...
2
votes
2answers
344 views

Asymptotic behaviour of sequence

I am interested in the sequence $$a(n)=\sum_{k=0}^n {p(n-k) \choose k}$$ where $p(n)$ is a polynomial equation. When $p(n)=n$ this reduces to the Fibonacci sequence, but what about when $p(n)$ is ...
3
votes
0answers
258 views

Generating function for the characteristic function of prime numbers

What do we know about the generating function of $\chi(n)$ (A010051) $$ f(x) = \sum_{n=0}^\infty \chi(n)x^n = \sum_{p\text{ prime}} x^p $$ for $\chi(n)$ the characteristic function of the primes: ...
1
vote
0answers
78 views

Rank generating functions on a graded poset and a linear extension of it

Let $A$ be a possibly infinite ensemble and let $\leq$ be a partial order between elements of $A$. We denote $P_{\leq}=(A,\leq)$ the corresponding poset. Furthermore, we suppose that $P$ is graded, ...