# Questions tagged [generating-functions]

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312
questions

2
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### Conjectural congruences for numbers related to Littlewood-Richardson coefficients

For $n \geq 0$, let $a_n$ be the square of the Euclidean length of the vector of Littlewood-Richardson coefficients of $\sum_{\lambda \vdash n} s_\lambda^2$, where $s_\lambda$ are the symmetric Schur ...

7
votes

0
answers

57
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### Property of an integer sequence related to series reversion

Thinking of some questions of homotopical algebra for operads, I ended up with a following question, perhaps someone will recognize something here:
Let $\{a_n\}_{n\ge 2}$ be a sequence of nonnegative ...

0
votes

0
answers

39
views

### String partition function

Hey I am a bit confused because when reading physics papers I encounter the free energy as a generating function of topological invariants as integrals over certain compactified moduli spaces. For ...

3
votes

1
answer

214
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### Analytic expression for the coefficient of a multivariate polynomial

Does there exist some method for finding an analytic expression for the coefficient of $z_1^kz_2^kz_3^k$ in:
$$[(1+z_1)(1+z_2)(1+z_3)(1+z_1z_2)(1+z_1z_3)(1+z_2z_3)(1+z_1z_2z_3)]^{k}$$
or is it ...

17
votes

3
answers

604
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### Matrices of combinatorial sequences that are inverse in two ways

I'm interested in pairs $A=(a_{i,j})_{i,j=0,1,\ldots}$ and $B=(b_{i,j})_{i,j=0,1,\ldots}$ of infinite matrices for which:
They are uni-lower-triangular, i.e., $a_{i,i}=1$ for all $i$ and $a_{i,j}=0$ ...

1
vote

3
answers

175
views

### Generating function of the square of Jacobi polynomial

The generating function of the Jacobi polynomials is given by
$$
\sum_{n=0}^{\infty} P_{n}^{(\alpha, \beta)}(z) t^{n}=2^{\alpha+\beta} R^{-1}(1-t+R)^{-\alpha}(1+t+R)^{-\beta}
$$
where
$$
R=R(z, t)=\...

3
votes

1
answer

213
views

### Generating function of the product of Legendre polynomials

The generating function of the product of Legendre polynomials for the same $n$ is given by
\begin{aligned}
\sum_{n=0}^{\infty} z^{n} \mathrm{P}_{n}(\cos \alpha) \mathrm{P}_{n}(\cos \beta)&=\frac{\...

0
votes

1
answer

164
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### Bounds on the number of integer compositions with parts bounded from above and below

I'm looking for asymptotic bounds (as n goes to infinity) on the number of integer compositions of $n$ with parts in $[a,n]$ and separately for parts in $[a,b]$, with $1 < a < b < n$.
(To ...

16
votes

6
answers

878
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### A Leibniz-like formula for $(f(x) \frac{d}{dx})^n f(x)$?

Let $f(x)$ be sufficiently regular (e.g. a smooth function or a formal power series in characteristic 0 etc.). In my research the following recursion made a surprising entrance
$$
f_1(x) = f(x),\ f_{n+...

6
votes

1
answer

315
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### How are Sheffer polynomials related to Lie theory?

Sheffer polynomials are orthogonal polynomial sets $\{P_n(x)\}$ having generating function $P(x,t) = \sum_{n=0}^{\infty}P_n(x)t^n=A(t)e^{xu(t)}$.
This form reminds me of the Lie group–Lie algebra ...

1
vote

0
answers

107
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### Hankel determinants of the sequence $(\binom{n}{m})_{n\ge0}$ and related sequences

I posted (https://math.stackexchange.com/questions/4363151/generating-functions-of-hankel-determinants-related-to-hoggatt-triangles) this question on Mathematics StackExchange but have not received a ...

6
votes

1
answer

238
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### Tanglegrams and functional equations of M. Somos

Recent references on the matter at hand include, a lecture slide The Konvalinka-Amdeberhan conjecture
and plethystic inverses and a preprint on Counting tanglegrams with species by I. Gessel; the ...

4
votes

1
answer

167
views

### Ratio of the first squared and the second moment

Let $G(t)$ be a probability generating function of some integer and non-negative r. v. $X$. Suppose that
$$\lim_{t\to1}G'(t)=+\infty.$$
That is
$$
\mathbb{E}X=+\infty.
$$
Can you show that
$$
\lim_{t\...

4
votes

0
answers

160
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### An addition theorem for three functions similar to $\sin,\cos$ and $\sinh,\cosh$ and one / some questions?

Define the functions $t_k(x) = \sum_{n=0}^{\infty}{\frac{x^{3n+k}}{(3n+k)!}}$ for $k=0,1,2$. The functions then satisfy:
$
\begin{pmatrix}\exp(x) \\ \exp(\omega x) \\ \exp(\omega^2 x)\end{pmatrix} =
\...

11
votes

1
answer

463
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### Generating function for Schur polynomials

Consider the generating function
$$
G_n(x_1,x_2,\ldots,x_n, t_1,t_2,\ldots,t_n) =\sum_{\lambda}s_{\lambda}(x_1,x_2,\ldots, x_n) t_1^{\lambda_1}t_2^{\lambda_2} \cdots t_n^{\lambda_n},
$$
where the sum ...

2
votes

0
answers

46
views

### Compact expression for triples of subsets with total sum zero

I am looking whether there is any compact way to write the following: Suppose we have an abelian group $G$. For a subset $A\subset G$ let $S_A$ be the sum of its elements. I want to find the number of ...

3
votes

2
answers

308
views

### Ask for a reference or a proof of a combinatorial identity $\sum_{k=0}^n\binom{2n+1}{2k}\binom {k}{m} =2^{2(n-m)}\frac{2n+1}{2(n-m)+1}\binom{2n-m}{m}$

Could you please recommend a reference to or supply a proof of the following identity \eqref{combin-ID-Maclaurin}, or \eqref{first-equiv-form}, or \eqref{combin-ID-Mac-Equiv}, or \eqref{combin-ID-Mac-...

4
votes

1
answer

179
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### Coefficients obtained from ratio with partition number generating function

This is a question inspired by T. Amdeberhan's recent question, as well as another previos MO question.
For an integer partition $\lambda$, and $k\in \mathbb{N}\cup\{\infty\}$, let $|\lambda|_k$ ...

2
votes

0
answers

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### Hilbert-function-like results for weighted projective spaces

Let $R = k[X_1,\ldots,X_N]/I$ be a finitely generated graded generated $k$-algebra, where the $X_i$'s have nonequal degrees, say $\deg(X_i)=a_i$, and $I$ is a (weighted-)homogeneous ideal.
We can ...

1
vote

1
answer

153
views

### A continued J fraction for $a_n = \frac{1}{(n+1)^2}$?

The following is called a J continued fraction:
$$\cfrac{\alpha_0}{1+a_0x-\cfrac{b_1x^2}{1+a_1x-\cfrac{b_2x^2}{1+a_2x-\cdots}}}$$
where the constants are real numbers. Let $\alpha_n= \frac{1}{(n+1)^2}$...

0
votes

0
answers

92
views

### A question on continued J-fraction

Consider the following two continued fractions $A$ and $B$:
$$\frac{\alpha_0}{1+a_0x-\frac{b_1x^2}{1+a_1x-\frac{b_2x^2}{1+a_2x-\cdots}}}$$
$$\frac{\beta_0}{1+c_0x-\frac{d_1x}{1+c_1x-\frac{d_2x}{1+c_2x-...

1
vote

1
answer

134
views

### Log-concavity of sequence related to overpartitions

The number $p_1(n)$ of overpartitions of $n$ is generated by
$$\sum_{n\geq0}p_1(n)\,q^n=\prod_{k=1}^{\infty}\frac{1+q^k}{1-q^k}.$$
Let $t\in\mathbb{N}$. Now, extend this to construct a family of ...

1
vote

0
answers

68
views

### Two-variable generating functions over coprime pairs

I am studying a sequence $(\alpha_{p,q})$ indexed by a pair of coprime integers; this sequence arises naturally in the study of a particular set of spaces in geometric topology, but unfortunately the ...

4
votes

1
answer

123
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### $0,1$-matrices with $1$ in every row/column vs. all $0,1$-matrices

Chapter 2, Exercise 25 of R. Stanley's "Enumerative Combinatorics" Vol. 1 asserts that
$$ \sum_{m,n \geq 0} \left(\sum_{t \geq 0} f_i(m,n)t^i\right)\frac{x^m}{m!}\frac{y^n}{n!} = e^{-x-y}\...

-1
votes

1
answer

117
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### Closed form for odd part of Bernoulli Polynomial generating function, $\sum_{k=0}^{\infty}B_{2k+1}(x)\frac{t^{2k+1}}{(2k+1)!}$ [closed]

If $B_k(x)$ are the Bernoulli polynomials, then (by definition, if you like) we get that
$$\sum_{k=0}^{\infty}B_k(x)\frac{t^k}{k!}=\frac{te^{tx}}{e^t-1}$$
My question is whether or not there is a ...

0
votes

1
answer

160
views

### Generating function for partial sums of the sequence

Let $p$ and $q$ be integers.
Let $f(n)$ be A007814, the exponent of the highest power of $2$ dividing $n$, a.k.a. the binary carry sequence, the ruler sequence, or the $2$-adic valuation of $n$.
Then ...

1
vote

1
answer

276
views

### What is the integral representation of the exponential function $e^{1/t}$ on $(0,\infty)$?

A function $q(x)$ is said to be completely monotonic on an interval $I$ if $q(x)$ has derivatives of all orders on $I$ and $(-1)^{n}q^{(n)}(x)\ge0$ for $x\in I$ and $n\ge0$. See Chapter 1 in the ...

20
votes

4
answers

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### Non-enumerative proof that, in average, less than 50% of tiles in domino tiling of 2-by-n rectangle are vertical?

Is there a non-enumerative proof that, in average, less than 50% of tiles in domino tiling of 2-by-n rectangle are vertical?
It is a nice exercise with rational generating functions (or equivalently, ...

3
votes

1
answer

246
views

### Generating functions for Hankel determinants of Catalan numbers

The Hankel determinants of the Catalan numbers are well known and can be written as
$d(k,n)= \det \left( C_{k + i + j} \right)_{i,j = 0}^{n - 1}=\prod_{i=1}^{k-1}\frac{\binom{2n+2j}{j}}{\binom{2j}{j}}$...

2
votes

0
answers

124
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### Generating Fuss-Catalan numbers using the regular Catalan number

Let $A_{n}(p,r)$ denote the $n$-th Fuss-Catalan number with parameter $(p,r)$. $A_{n}(p,r)$ has the closed form
$A_{n}(p,r) = \frac{r}{np+r} {np+r \choose n}.$
For example,
$A_{n}(2,1) = \frac{1}{n+1} ...

2
votes

0
answers

68
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### Finite version of Mehlers formula?

This is a crosspost from Math Stack Exchange, please let me know if this is not an appropriate use of crossposting, and I will delete.
Mehler's formula is the following identity for Hermite ...

3
votes

0
answers

100
views

### Irreducible dimensions generating function for Lie algebra $\mathfrak{sl}_n$

Let $\lambda = \sum_{i = 1}^{n - 1} m_i \omega_i$ be the highest weight of irreducible representation $V(\lambda)$ of Lie algebra $\mathfrak{sl}_n$. As we know from the Weyl formula,
$$\dim V(\lambda) ...

3
votes

0
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118
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### $q$-series for the number of rectangles in a square lattice

Given a partition $\lambda\vdash n$ of $n$, look at its Young diagram $Y_{\lambda}$. Let $a(\lambda)$ be the number of squares (of all sizes) in $Y_{\lambda}$. For example, if $n=4$ then $a(4)=4, a(3,...

3
votes

1
answer

238
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### A generating function related to the Delannoy numbers

What is the generating function of $f_{m,n}$?
$ f_{m,n} = \begin{cases} 0 , & \text{if $m<0 $ or $ n<0$ }; \\
f_{n,m} , & \text{ if $n<m$}; \\
1, & \text{ if $0=m$ and $ n\...

1
vote

0
answers

143
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### Asymptotics of $\sum \frac{d(n)}{n}$ with generating functions

We can determine the asymptotics of partial sums involving the divisor function accurately by means of, for example, the hyperbola method: $$\sum_{n\leq N}\frac{d(n)}{n}=\frac{1}{2}(\log(N))^{2}+2\...

3
votes

0
answers

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### Positivity of sequences

Totally positive sequences $\lbrace a_n\rbrace_{n\in\mathbb{Z}}$ are defined as those such that the Töplitz matrix $A_{ij}=a_{i-j}$ is totally positive (all its minors are non-negative). An ...

0
votes

1
answer

201
views

### Product of three or more independent sub-Gaussian varibles

A random variable $X$ is called subgaussian of order $\sigma^2$ if $\log E[exp\{\theta X\}]\leq \frac{1}{2}\theta^2\sigma^2$ for every $\theta\in\mathbb R$.
Given a sequence of independent subgaussian ...

0
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0
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145
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### Asymptotically similar functions with opposite parity, were they considered, are they useful? Case of polynomials

So, can we transform an even function into an odd function and vice versa?
Let's consider this method:
Transformation even->odd:
Suppose $f_{even}(x)$ is a function which satisfies the following ...

2
votes

1
answer

257
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### Radius of convergence of cumulant generating function

Recall that for a random variable $X$ with a moment generating function $M_X(t)$ the cumulant generating function is defined as
\begin{align}
K_X(t)=\log M_X(t)
\end{align}
The Taylor expansion of $...

3
votes

2
answers

341
views

### Number of ways of distributing indistinguishable balls into distinguishable boxes with extra givens

What is the number of ways to distribute $m$ indistinguishable balls to $k$ distinguishable boxes given no box can be a unique number of balls?
for example: ($m=19$ and $k=5$)
$$x_1 + x_2 + \dots + ...

0
votes

1
answer

123
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### Is there a closed form of $\sum_{i=1}^{n-k} {n-1-i\choose k-1}x^i$ in $x$?

I'm looking for the generating function of the sum $\sum_{i=1}^{n-k} {n-1-i\choose k-1}x^i$. One can compute this using the Euler-MacLauren formula but the remainder term is a little messy. Is there ...

0
votes

1
answer

133
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### How to probe the recursiveness order of a sequence $\{S_n\}$ whose generating function is known

How to probe the recursiveness order of a sequence $\{S_n\}$ whose generating function is known:
$$ \sum_{n\geq0} S_n z^n= \frac{4 z \left(\sqrt{49 z^2-18 z+1}+7 z-1\right)}{\sqrt{49
z^2-18 z+1} \...

10
votes

1
answer

256
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### Software for recognizing algebraic or D-finite formal power series

I have a formal power series in one variable that I think might be algebraic (or perhaps just D-finite). Is there software that could help me explore this?
By way of comparison, there’s a very simple ...

13
votes

2
answers

325
views

### What alternatives are there to the binomial poset theory of generating function families?

A natural question in combinatorics is, why are certain families of generating functions combinatorially useful, like $\Sigma_n a_n x^n$ and $\Sigma_na_n\frac{x^n}{n!}$, why are other families are not,...

3
votes

1
answer

412
views

### Conjecture on bernoulli numbers and binomial coefficients

Crossposted from
https://math.stackexchange.com/questions/4116414/conjecture-on-bernoulli-numbers-and-binomial-coefficients
In playing around with some formulas, I have come up with the following ...

2
votes

1
answer

163
views

### Fibonacci-Motzkin paths and J-type continued fractions

Recall that a Motzkin path is a piece-wise linear planar path
connecting points in the integer lattice quadrant
$\Bbb{Z}_{\geq 0} \times \Bbb{Z}_{\geq 0}$ beginning at the origin $(0,0)$ and
ending at ...

1
vote

1
answer

294
views

### Why $\lim_{n\rightarrow \infty}\frac{F(n,n)}{F(n-1,n-1)} =\frac{9}{8}$?

$$F(m,n)= \begin{cases}
1, & \text{if $m n=0$ }; \\
\frac{1}{2} F(m ,n-1) + \frac{1}{3} F(m-1,n )+ \frac{1}{4} F(m-1,n-1), & \text{ if $m n>0$. }%
\end{cases}$$
Please a proof of:
$$\lim_{...

1
vote

1
answer

678
views

### What is the inverse Laplace transform of $\frac{(1/s)_{n}}{s} $?

Introduction
So far, I have found (p. 5) the following generating functions of the unsigned Stirling numbers of the first kind:
\begin{equation} \tag{1} \label{1} \sum_{l=1}^{n} |S_{1}(n,l)|z^{l} = (z)...

12
votes

2
answers

569
views

### Alternating sum of hook lengths: Part I

Given $\lambda$ an integer partition of $n$, let $h_{ij}(\lambda)$ denote the hook length of cell $(i,j)$ in the Young diagram of $\lambda$.
Is there a closed formula or a generating function for the ...

1
vote

0
answers

36
views

### Correspondence between monomer-dimer heaps and words in 2 alphabets

For background and some illustrative pictures, refer to this preprint by A M Grasia and G Ganzberger: Fibonacci polynomials. For the present purpose, it suffices to read into pages 4 and 5.
The part ...