Questions tagged [generating-functions]

A generating function is a way of encoding an infinite sequence of numbers by treating them as the coefficients of a formal power series. Tag questions involving generating functions in this.

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Moment generating function for product states

In the sequel $B=M_\ell(\mathbb{C})$. For $M\in\mathbb{N}$ fixed and $N\geq M$ I consider the symmetrizer $\pi_{M,N}(x_M)\in B^{\otimes N}$, which is the symmetrized tensor product of $a_1$,...,$a_M$ ...
Kris's user avatar
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5 votes
1 answer
110 views

Identities for the generating functions of a sort of convolution powers of the Narayana numbers

Let $c(x)=\frac{1-\sqrt{1-4x}}{2x}$ be the generating function of the Catalan numbers. It satisfies $$\frac{1}{c(x)^k}+x^k c(x)^k=L_k(1,-x),$$ where $L_n(x,s)$ denote the Lucas polynomials defined by $...
Johann Cigler's user avatar
2 votes
1 answer
238 views

The probability that iid draws from a mean zero random variable sum to zero

Suppose we have a probability distribution $p(\cdot)$ supported on the integers between $-m$ and $m$ for some positive integer $m$, with $\sum_k kp(k) = 0$. Suppose furthermore that all $p(k)$ are ...
James Propp's user avatar
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Recurrence involving a convolution-like sum

During a research we are finding several sequences $g_n$ which are defined by recurrences of the form: $$g_n=\sum_{k=1}^{n-1} a_{k,n} g_k g_{n-k}$$ For some sequence $a_{n,k}$ (usually defined by a ...
user3141592's user avatar
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44 views

$R$-recursion for the A007165

Let $a(n)$ be A007165 i.e. number of $P$-graphs with $2n$ edges. Here ordinary generating function $A(x)$ satisfies $$ A(x) = \frac{(1 + xA(x))(1 + 2xA(x))}{1 + 2xA(x) - (xA(x))^2} $$ Let $$ R(n, q) = ...
Notamathematician's user avatar
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48 views

$R$-recursion for the A036765

Let $a(n)$ be A036765 i.e. number of ordered rooted trees with $n$ non-root nodes and all outdegrees $\leqslant 3$. Here $$ a(n) = \frac{1}{n+1}\sum\limits_{j=0}^{\left\lfloor\frac{n}{2}\right\rfloor}\...
Notamathematician's user avatar
1 vote
1 answer
89 views

General case of the some $R$-recursions

Let $f(n)$ be an arbitrary function. Let $a(n)$ be an integer sequence such that its ordinary generating function satisfies $$ A(x)=\sum\limits_{i=0}^{\infty}\frac{x^i}{\prod\limits_{j=0}^{i}(1-f(j)x)...
Notamathematician's user avatar
1 vote
1 answer
84 views

$R$-recursion for the A307389

Let $a(n)$ be A307389 i.e. an integer sequence such that its exponential generating function $A(x)$ satisfies $$ A(x)=\exp\left(\frac{\exp(2x)-2\exp(x)+2x+1}{2}\right) $$ The sequence begins with $$ 1,...
Notamathematician's user avatar
3 votes
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$R$-recursion for the A249833 (similar to A235129)

Let $a(n)$ be A249833 i.e. an integer sequence such that its exponential generating function $A(x)$ satisfies $$ A(x) = 1 + \int A(x) + (A(x))^2\log A(x)\,dx $$ The sequence begins with $$ 1, 1, 2, 7, ...
Notamathematician's user avatar
2 votes
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102 views

$R$-recursion for the A235129

Let $a(n)$ be A235129 i.e. an integer sequence such that its exponential generating function $A(x)$ satisfies $$ A'(x) = 1 + A(x)\exp(A(x)) $$ The sequence begins with $$ 1, 1, 3, 12, 64, 424, 3358, ...
Notamathematician's user avatar
3 votes
2 answers
171 views

How to diagonalize this tridiagonal difference operator with unbounded coefficients?

Problem: I have a self-adjoint operator in $\ell^2(\mathbb{Z})$ which acts as $$T g(x)=q^{-2 x -3/2} g(x+1)+(1+q) q^{-2 x-1} g(x)+q^{-2 x +1/2} g(x-1),$$ and I am looking to diagonalize it. The ...
Leonid Petrov's user avatar
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1 answer
152 views

A self-consistent equation that turns into a differential equation

Suppose the function $f(x,y)$ is defined on a small neighbourhood of $(0,0)$ in $\mathbb{R} \times [0,\infty)$ and satisfies the self consistent equation \begin{align*} & f(x,y) = \frac{1}{1-y} + ...
Tardis's user avatar
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2 votes
1 answer
198 views

Generating function over primes in an arithmetic progression

Given a newform $\sum_{n=1}^{\infty}a(n)q^n$. Is the generating function $$ \sum_{p\equiv a\pmod{m}}a(p)q^p $$ over the primes $p\equiv a\pmod{m}$ still a modular form? Any help is highly appreciated! ...
ModularForms's user avatar
1 vote
2 answers
233 views

Recurrence relation with two variables

I am stuck on a recurrence relation with two variables. I'm familiar with techniques to solve recurrence relations with one variable and looked into ways to solve recurrence relations with multiple ...
user675763's user avatar
2 votes
1 answer
185 views

Slicing bivariate exponential generating functions on x and y

Let $F(x, y) = e^{y D(x)}$ be a generating function for sets of objects enumerated by $D(x)$ that also keeps track of the number of sets (enumerated by the variable $y$, while $x$ enumerates the total ...
Oleksandr  Kulkov's user avatar
1 vote
2 answers
343 views

A closed formula for a sum involving hypergeometric function

Let ${ }_1 F_1(a ; c ; z)$ be Kummer's function defined by the function, and all its analytic continuations, represented by the infinite series $\sum_{n=0}^{\infty} \frac{(a)_n}{(c)_n} \frac{z^n}{n !...
zoran  Vicovic's user avatar
19 votes
2 answers
699 views

A rational function related to Fibonacci numbers

Let $F_n$ denote a Fibonacci number ($F_1=F_2=1$, $F_{n+1}=F_n+F_{n-1}$ for $n\geq 2$). Define $$\prod_{k=1}^n (1+x^{F_{k+1}}) = \sum_j f(n,j)x^j. $$ For a positive integer $r$ let $$ v_r(n) = \sum_j ...
Richard Stanley's user avatar
7 votes
0 answers
264 views

A conjecture about Hankel determinants of path generating functions

Let $a_{n,k}=a_{n,k}(x,c)$ be the generating function $\sum_P w(P),$ where $P$ runs over all paths from $(0,0)$ to $(n,k)$ consisting of horizontal steps $(1,0)$, up-steps $(1,1)$ and down-steps $(1,-...
Johann Cigler's user avatar
1 vote
0 answers
74 views

How to calculate the Integral with confluent hypergeometric function

How to prove this.Thank you in advance Let $\delta,\beta>0$ How to prove this \begin{align} & \int^1_0 \frac{w^{1-\beta}}{(1-w)^{1+\delta}} (-t.s w)^{\frac{-\delta}{2}} e^{-\frac{w}{1-w}(s+t)}...
zoran  Vicovic's user avatar
7 votes
1 answer
474 views

Combinatorial consequences of de Branges's Theorem?

I'm usually not a proponent of the mentality “here is a tool, what results can we prove with it?” (as I prefer to start at the other end with a well-motivated problem), but this famously entertaining ...
Erik Lundberg's user avatar
3 votes
0 answers
208 views

Number of partitions of set restricted by sum of square of part size

Let $p_1^{a_1}p_2^{a_2}\cdots$ denotes the integer partition of $n$, i.e. $a_1p_1+a_2p_2+\cdots=n$. Or equivalently $m_1+m_2+\cdots=n$. It is known that the number of partitions of set $\{x_1,x_2,\...
tony's user avatar
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2 votes
1 answer
330 views

Combinatorial meaning of a binomial expansion

Let $F$ be a generating function $F(x) = \sum_{i=0}^\infty f_i x^i$, and suppose that we can do operations formally without worrying about convergence issues. Define the coefficients \begin{gather*} ...
Student's user avatar
  • 5,008
4 votes
0 answers
158 views

Sum $f(n_1,n_2,\ldots,n_k) 1^{n_1} 2^{n_2} \ldots k^{n_k}$ over partitions

Use the notation $(n_1,n_2,\ldots,n_k) \vdash n$ to denote that $(n_1,n_2,\ldots,n_k)$ is a partition of the positive integer $n$, that is, $n_1+n_2+\ldots+n_k = n$ and $n_1 \ge n_2 \ge \ldots \ge n_k ...
Dreamer's user avatar
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3 votes
1 answer
151 views

$q$-series and Stirling of the 1st kind

Denote the (unsigned) Stirling numbers of the $1^{st}$-kind by ${n \brack k}$ and define $$\mathbf{F}_a(q)=\sum_{m\geq1}\frac{q^{am}}{(1-q^m)^{2a}} \qquad \text{and} \qquad \mathbf{G}_b(q)=\sum_{m\...
T. Amdeberhan's user avatar
1 vote
0 answers
85 views

Suitable recursion for the A234289

Let $a(n)$ be A234289 i.e. integer sequence with exponential generating function $$ A(x)=1+A(x)^2\int \frac{1}{A(x)}\,dx $$ The sequence begins with $$ 1, 1, 3, 17, 147, 1729, 25827, 468593, 10012083, ...
Notamathematician's user avatar
1 vote
2 answers
150 views

Transcendental functions with two prescribed values

Let $\alpha$ and $\beta$ two algebraic numbers lying in unit ball. Let $T:=(t_k)_k$ be an increasing sequence of positive integers such that $t_{k+1}/t_k$ tends to $1$ as $k\to \infty$. I would like ...
Jean's user avatar
  • 515
1 vote
0 answers
77 views

Recursion for the A006014 using difference of binomial coefficients

Let $a(n)$ be A006014 i.e. $$ a(n)=na(n-1)+\sum\limits_{j=1}^{n-2}a(j)a(n-j-1), \\ a(1)=1 $$ Also generating function $A(x)$ satisfies $$ A(x) = x(1 + A(x) + A(x)^2 + xA'(x)) $$ Let $$ R(n,q)=\sum\...
Notamathematician's user avatar
0 votes
0 answers
67 views

Recursion for a given series reversion

Define the operator $\operatorname{SR}$, which is associated with the series reversion. Let $a(n,m,k)$ be an integer sequence with generating function $$ \frac{1}{x}\operatorname{SR}(x\frac{1-mx}{1-kx}...
Notamathematician's user avatar
4 votes
0 answers
117 views

Something (which might be called multi-continued fraction) for the A112487

Let $a(n)$ be A112487 i.e. an integer sequence with exponential generating function $$ A(x)=\exp\left(\int (A(x)+A(x)^2)\,dx\right), \\ A(0)=1 $$ However, the definition in the name of the sequence is ...
Notamathematician's user avatar
0 votes
0 answers
100 views

Recursion for the A266328 by analogy with non-standard recursion for factorials

Let $a(n)$ be A266328 i.e. an integer sequence with exponential generating function $$ A(x)=\exp\int B(x) \,dx $$ such that $$ B(x)=\exp(-x)\exp\int A(x) \,dx $$ where the constant of integration is ...
Notamathematician's user avatar
0 votes
0 answers
78 views

Simple recursion for the A129179

Let $T(n,k)$ be A129179, i.e., an integer coefficient with generating function $$ G(t,z) = 1 + zG(t,z) + tzG(t,t^2z)G(t,z) $$ Other generating functions are $\frac{1}{G_1(t,z,0)}$ and $\frac{1}{G_2(t,...
Notamathematician's user avatar
0 votes
2 answers
260 views

Simplification of hypergeometric Function

First of all I am not at all a math expert, but I have some working knowledge. That said, please excuse "dumb" questions. I am looking at the following process: Assume you are on the 2-...
WaveL's user avatar
  • 9
0 votes
0 answers
168 views

Expansion of continued fraction using recursion

Let $f(n)$ be an arbitrary function such that $f(n)\in\mathbb{Z}$. Let $a(n)$ be an integer sequence with generating function $\frac{1}{G(0)}$ where $$ G(j)=1-\frac{f(j)x}{G(j+1)} $$ Here we have $$ G(...
Notamathematician's user avatar
0 votes
0 answers
63 views

General patterns for partial sums of generalized A341392, A284005 and A329369

Let $$\ell(n)=\left\lfloor\log_2 n\right\rfloor$$ $$f(n)=n-2^{\ell(n)}$$ $$\operatorname{wt}(2n+1)=\operatorname{wt}(n)+1, \operatorname{wt}(2n)=\operatorname{wt}(n), \operatorname{wt}(0)=0$$ $$T(n,k)=...
Notamathematician's user avatar
4 votes
0 answers
200 views

Extract this constant term

Given a Laurent polynomial $F$ in the variables $\mathbf{t}=(t_1,\dots,t_n)$, let $CT_{\vec{\mathbf{t}}}\,F$ denote its constant term. For example, $CT_{t_1,t_2}((8t_1-\frac1{3t_1t_2})(5t_1t_2+t_2^2+\...
T. Amdeberhan's user avatar
1 vote
1 answer
331 views

Products involving exponents of tribonacci numbers

The Fibonacci numbers $F_n$ can be given by $$\sum_{k\geq0}F_kx^k=\frac{x}{1-x-x^2}.$$ Among many many properties of this sequence, consider the following two results: (1) the coefficients of the ...
T. Amdeberhan's user avatar
0 votes
0 answers
71 views

Sequences that sum up to possible generalization of Euler or up/down numbers (A000111)

Let $a(n,m,k)$ be an integer sequence with e.g.f. $$A(x)=\operatorname{exp}\left(x + m\int\int (A(x))^k \, dx \, dx\right)$$ I don't know much about integrals, so here's a concrete example: $a(n,1,3)$...
Notamathematician's user avatar
1 vote
0 answers
85 views

Application of the series reversion

Let $f(n)$ be an arbitrary function such that $f(n)\in\mathbb{Z}$. Let $a(n)$ be an arbitrary integer sequence such that $a(0)=1$. Let $b(n)$ be an integer sequence such that $$b(2^m(2n+1))=\sum\...
Notamathematician's user avatar
0 votes
1 answer
186 views

Fibonacci and product polynomials

The motivation for my current question arises from this MO post by R. Stanley. Caveat. There's a slight alteration. With the convention $F_1=F_2=1$ for the Fibonacci numbers, define the polynomials $...
T. Amdeberhan's user avatar
0 votes
0 answers
35 views

Representation theorem for multivariate homogeneous linear recurrences on Z^d?

Let $f:\mathbb{Z}^d \to \mathbb{C}$ satisfy a homogeneous linear recurrence for some coefficients $a_\Delta \in \mathbb{C}$, $$\forall x \in \mathbb{Z}^d. \quad \sum_{\Delta \in B_k(0)}a_\Delta f(x+\...
Chris Jones's user avatar
1 vote
2 answers
294 views

Closed formula for Hermite polynomials

Hermite polynomials $H_k(x), x \in \mathbb{R}, k \in \mathbb{N}$ are defined by the formula $$ H_k(x)=(-1)^k e^{x^2} \frac{d^k}{d x^k}\left(e^{-x^2}\right) . $$ Each $H_k(x)$ is a polynomial of exact ...
zoran  Vicovic's user avatar
1 vote
1 answer
120 views

How to interpret this result modulo $(y-1)^{n+1}$?

I recently discovered that the following identity is true: $$ \boxed{\frac{\partial^{n+1}}{\partial x^{n+1}}\left(\frac{(xy-1)^n}{n!} \log \frac{1}{1-x}\right) \equiv \frac{y^{n+1}}{1-xy} \pmod{(y-1)^{...
Oleksandr  Kulkov's user avatar
0 votes
0 answers
70 views

When is the logarithmic generating function of relative compositions negative at −1?

Suppose $f\colon \mathbb{N} \to \mathbb{R}$. Define the logarithmic generating function of $f$ to be $$ L_{f}(x) = \sum_{k = 1}^\infty f(k) \frac{x^k}{k}. $$ This is in contrast to the exponential ...
Nate Ackerman's user avatar
1 vote
0 answers
53 views

Over a given finite field, how many couples of matrices there are, for which their minimal polynomials are co-prime?

Let ${\mathbb F}_{q}$ be a given finite field. How many couples of $n\times n$ matrices $\left(A,B\right)$ over ${\mathbb F}_{q}$, such that $\gcd\left(\mu_{A}\left(\lambda\right),\mu_{B}\left(\lambda\...
Yossi Peretz's user avatar
8 votes
1 answer
252 views

Use of generating functions in logic

Are there any uses of generating functions within logic, in particular to count how many models exists for a given theory $T$, say in FOL? The concrete problem I'm hoping to apply this to involves ...
Steven Schaefer's user avatar
3 votes
1 answer
259 views

name for products of the form $\prod_i (1 + a_i t^i)$

In the context of generating functions, is there an established name for (infinite) products of the form $\prod_i (1+a_it^i)$, or perhaps more generally $\prod_i (1+f_i(t))$, assuming that the ...
Martin Rubey's user avatar
  • 5,503
4 votes
0 answers
391 views

Explicit formula for tournament sequence

I am looking for an explicit formula for a sequence. The sequence is generated as follows: There is a tournament with $10$ teams. In the beginning, all teams have a 0-0 win-loss record. The teams are ...
Jackson's user avatar
  • 41
2 votes
0 answers
109 views

Asking for a generating function for an arithmetic sequence

For fixed integer $n\geq1$, let $c_m(n)$ be the number of divisors $d$ of $m$ such that $n<d\leq 2n$. Here is an experimental generating function for which I ask: QUESTION. Is this true? $$\sum_{m\...
T. Amdeberhan's user avatar
0 votes
0 answers
83 views

Arithmetic triangles and unimodality of its rows

Let's consider the sequence of coefficients of $\prod_{i}\frac {1-x^{d_i}} {1-x}$, where $d_i$ is a monotonically increasing nonnegative integer sequence. How to prove that the coefficients form an ...
Mikhail Gaichenkov's user avatar
6 votes
0 answers
252 views

Usefulness of total algebras and exotic generating series

In his first Algebra volume, Bourbaki [1] defines the structure of a “total algebra” i.e. the space of functions on a monoid $M$ (to a ring $k$) with the convolution product ( a function $f:\ M\to k$ ...
Duchamp Gérard H. E.'s user avatar

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