# 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.

385
questions

4
votes

1
answer

133
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### Inverse relationship between Stirling numbers of the first and second kind via generating functions

In combinatorics, a well-known result is that the matrix formed by the Stirling numbers of the second kind $\left(S(n,k)\right)_{n,k\geq 0}$ and the matrix of the signed Stirling numbers of the first ...

7
votes

0
answers

108
views

### A differential equation and recurrence related to P-partitions

I am interested in polynomials $G_n(z)$ defined by the recurrence
$$G_{n+1}(z) - 2G_n(z) + (1-nz)G_{n-1}(z)=0$$
for $n\ge1$ with the initial values $G_0(z) = 1$ and $G_1(z) = 1$.
The next few values ...

2
votes

2
answers

253
views

### 5 different ways to define the same family of integer sequences

Let ${n \brace k}$ be a Stirling number of the second kind.
Let $A_n(x)$ be an Eulerian polynomial. Here
$$
A_n(x) = \sum_{i=0}^{n}i!{n \brace i}(x-1)^{n-i}.
$$
Let $a_1(n,p,q)$ be the family of ...

3
votes

1
answer

117
views

### $R$-recursion for unsigned Genocchi numbers (of first kind) of even index

Let $G_n$ be A036968 (i.e., Genocchi numbers). Here
$$
\frac{2t}{1+e^t}=\sum\limits_{n=0}^{\infty}G_n\frac{t^n}{n!}.
$$
Also
$$
t\tan\left(\frac{t}{2}\right)=\sum\limits_{n=1}^{\infty}(-1)^n G_{2n}\...

9
votes

1
answer

488
views

### Solving a second-order recurrence relation / Series expansion of a confluent Heun equation

I would like to know whether it is possible to solve (in "closed form") either one of the following two second-order recurrence relations, which are closely related to each other. The first ...

22
votes

2
answers

2k
views

### Coincidence between coefficients of tanh(tan(x/2)) and Chow ring computations?

In "Kodaira–Spencer Theory of Gravity and Exact Results for Quantum String Amplitudes" by Bershadsky, Cecotti, Ooguri, and Vafa (arxiv) on pg. 96 appear the two numbers $5760$ and $1451520$ ...

2
votes

1
answer

471
views

### Generalized Bernoulli numbers

In Euler–Maclaurin formula Bernoulli numbers express a finite sum through the integral. In my generalization a finite sum is expressed through another finite sum with a different step. All that is ...

1
vote

1
answer

106
views

### Closed formula / asymptotics for a generating function involving Gegenbauer / ultraspherical polynomials

Are there asymptotics, or even a closed form, for the following series
$$ \sum_{k = 0}^\infty e^{2 \pi i \sqrt{k^2 + (d-1) k} } \left( \binom{d+k}{k} - \binom{d+k-2}{k-2} \right) G_{\frac{d-1}{2},k}(t)...

2
votes

2
answers

203
views

### Monotonicity of the sum of coefficients of a family of generating functions

Let
\begin{equation*}
A_{n,w}(z)=\left(\sum_{i=0}^{\lfloor\frac{w}{2}\rfloor-1}\binom{w}{i}z^i+\frac{1}{2^{(w+1)\bmod 2}}\binom{w}{\lfloor\frac{w}{2}\rfloor}z^{\lfloor\frac{w}{2}\rfloor}\right)^{n/w}
\...

0
votes

0
answers

31
views

### Convergence region of multivariate rational functions

Assume $p, q \in \mathbb{R}[x_1,\ldots,x_k]$ and let $ \vec{0} \not\in V(q) := \{\vec{x} \in \mathbb{R}^k \mid q(\vec{x}) = 0 \}$ such that $r_q := \inf_{\vec{x}\in V(q)} |\!|\vec{x}|\!|_\infty < \...

0
votes

0
answers

44
views

### Counting monotonic arrays

Let $\mathcal M_d(N)$ denote the collection of functions from $\mathbb N_{0}^d\to\mathbb N_0$ with the following properties:
$f(\mathbf n+\mathbf e_i)\le f(\mathbf n)$ for all $\mathbf n\in \mathbb ...

5
votes

1
answer

224
views

### Reference request: Gessel interview's generating function identities

In this interview, Ira Gessel mentions the following results:
Result 1: Let $B_n$ denote the $n^{\text{th}}$ Bernoulli number.
Define the series
$$B(x) = \sum_{n=2}^{\infty} \frac{B_nx^{n-1}}{n(n-1)}.$...

0
votes

0
answers

50
views

### Solving non-linear recursion with linear and exponential terms

I encountered the recursion $$\frac{a[n+2]}{a[n+1]}-e^{-\frac{a[n+1]}{a[n]}}=0$$ when trying to explain why points are apparently arranged along an exponential curve in the scatter plot of an empiric ...

2
votes

5
answers

531
views

### Finding $\sum_i x_i$ given $\{\sum_i x_i^{2n}\}_{n\in \mathbb{N}}$

Can we find $\sum_i x_i$ given $\{\sum_i x_i^{2n}\}_{n\in \mathbb{N}}$, assuming $\{x_i\}_{i\in\mathbb{N}}$ is a set of positive real numbers?
Perhaps an easier question is, can we find $\sum_i x_i$ ...

2
votes

1
answer

153
views

### Weak compositions with no subcomposition adding to (more than) $j$

Here is a solution to a problem from Stanley's Enumerative Combinatorics (it's listed as a difficulty 2, so I imagine what I'm about to ask is likely a 2+ or 3-) about the number $\kappa(N,k,j)$ of ...

9
votes

2
answers

321
views

### Counting $m\times n$ $\bigl({1\atop1}{1\atop0}\bigr)$-free $(0,1)$-matrices

Let $G_{m,n}$ denote the number of $m\times n$ $(0,1)$-matrices that avoid the submatrix $\bigl({1\atop1}{1\atop0}\bigr)$. (Submatrices need not be contiguous.) Here are some small values (not yet on ...

0
votes

0
answers

29
views

### 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 $a_1$,...,$a_M$ ...

5
votes

1
answer

134
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 $...

2
votes

1
answer

252
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 ...

0
votes

0
answers

48
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) = ...

1
vote

0
answers

49
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}\...

1
vote

1
answer

95
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)...

1
vote

1
answer

89
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,...

3
votes

0
answers

70
views

### $R$-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, ...

2
votes

0
answers

103
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, ...

4
votes

2
answers

183
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 ...

0
votes

1
answer

168
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} + ...

2
votes

1
answer

218
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! ...

1
vote

2
answers

319
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 ...

2
votes

1
answer

201
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 ...

1
vote

2
answers

370
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 !...

20
votes

2
answers

721
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 ...

7
votes

0
answers

275
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,-...

1
vote

0
answers

77
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)}...

7
votes

1
answer

487
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 ...

3
votes

0
answers

217
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,\...

2
votes

1
answer

331
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*}
...

4
votes

0
answers

211
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 ...

3
votes

1
answer

154
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\...

1
vote

0
answers

89
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, ...

1
vote

2
answers

159
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 ...

1
vote

0
answers

80
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\...

4
votes

0
answers

118
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
...

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 ...

0
votes

2
answers

316
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-...

0
votes

0
answers

175
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(...

4
votes

0
answers

207
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+\...

1
vote

1
answer

337
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 ...

0
votes

0
answers

73
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:
$a(n,1,3)$...

1
vote

0
answers

92
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\...