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Questions tagged [generating-functions]

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17 views

The asymptotics of a vector sequence defined by a recursion relation

The sequence of vectors $(\mathbf{v}_0,\mathbf{v}_1,\mathbf{v}_2,\dots)$ obeys the recursion relation that $A\mathbf{v}_j-\mathbf{v}_{j-1}=\sum_{k=0}^j diag(\mathbf{v}_k)B\mathbf{v}_{j-k}$, where A ...
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0answers
71 views

One-point partition

Let following be the partition function in infinitely many variables $x_i$ the linear coordinates on the vector space $V$. $$ \mathcal{Z}=exp\Big(\sum_{\substack{g\geq 0\\n\geq 1}}\frac{h^{g-1}}{n!}\...
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0answers
103 views

Exponential generating function for k^n / n

The ordinary generating function for $k^n/n$ is $-\log(1-kx)$. Is there a closed-form exponential generating function for this sequence?
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0answers
99 views

Relate log concavity of sequence to convexity of 1/generating function

Let $Y$ be a random variable on the natural numbers, with $p_k = \mathbb{P}(Y = k)$ for $k \geq 0$. Say that $Y$ is log-concave if $p_k^2 \geq p_{k-1} p_{k+1}$ for $k \geq 1$, and that $Y$ is log-...
5
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1answer
153 views

Collapsed partitions and generating functions

Given $n\in\Bbb{N}$, the number of (unrestricted) integer partitions of $n$ are given by $$\sum_{n\geq0}p(n)x^n=\prod_{j\geq1}\frac1{1-x^j}.$$ Define the collapsed partitions of $n$ to be the ...
1
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1answer
120 views

Integer partitions with subset sums “not divisible” by p

I have the following questions: Let $N \in \mathbb{N}$ and \begin{equation} \sum_{i=1}^k n_i = N, \end{equation} with $n_i \in \mathbb{N}$ for $1 \le i \le k$ and some $k \in \mathbb{N}$, be an ...
2
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2answers
292 views

Explicit formula for a generating function

In an enumeration problem the sequence of number of Dyck paths semilength n having no UUDD's starting at level 0 with generating function $$\frac{2}{(1+2z^2+\sqrt{1-4z})}$$ showed up, see also https://...
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0answers
43 views

About basis of function spaces under composition

Say $L^2(M_1,\mathbb{R})$ is the space of square-integrable functions on $M_1$ which is a compact manifold with a measure defined on it. Let $L^2(\mathbb{R}^n,\mathbb{R})$ be the space of all square-...
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0answers
55 views

Generating function for number of r-disjoint subsets each of size k

Fix $n, k$. Let $$ C^{n,k}_r =\frac{1}{r!} \binom{n}{\underbrace{k, \ldots, k}_{\text{r times}}, n-rk} = \frac{n!}{r!(k!)^r(n - kr)!} $$ be the number of ways to form $r$ disjoint subsets each of ...
17
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1answer
636 views

Congruences Ramanujan-style

Let $t\in\Bbb{N}$ and consider the sequences $p_t(n)$ defined by $$\sum_{n\geq0}p_t(n)x^n=\prod_{i\geq1}\frac1{(1-x^i)^t}=(x;x)_{\infty}^{-t}.$$ The numbers $p_t(n)$ can be regarded as enumerating ...
3
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2answers
267 views

Seeking proof to an asymptotics of a recursion or functional equation

My question on math.stackexchange.com and the continuation by an answer to it gives the two summation expressions for the recursion $$a_n = 1+\frac1{2^n}\sum_{k=0}^n {n\choose k}a_k,\, \forall n\in\...
4
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1answer
163 views

Second order recurrence relation for third order polynomial root

Consider this recurrence relation: $$ \begin{eqnarray*} f_0&=&1\\ f_n&=& \sum_{m=0}^{n-1} \frac{\left(\frac{m+3}{2}\right)_{m-1}}{\left(\frac{m+2}{2}\right)_m} f_{n-m-1} f_m\ \ \ \...
3
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0answers
85 views

How to use this generalised 'generating function' for the Gegenbauer polynomials

Cohl (2011) gives a generalisation of the standard generating function for the Gegenbauer polynomials $C_n^\mu(x)$: $(1 -2tx + t^2)^{-\nu} = A_{\mu,\nu} \frac{(1-t^2)^{-\nu+\mu+1/2}}{t^{\mu+1/2}} \...
10
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0answers
291 views

Wherefore art thou a Borcherds Product?

This question essentially asks how can one recognize (or rule out) that a generating function of combinatorial origin may be given as a Borcherds type product. I'll start with a motivational example: ...
0
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1answer
144 views

Is this a Borel summable $ S = \sum_{k=0}^\infty (-1)^k (k!)a_k $ with $ a_k$ alternating sequence?

let $ S = \sum_{k=0}^\infty (-1)^k (k!)a_k $ a divergent series such that $b_k=(-1)^k (k!)a_k >0 $ for $k>1$ , and $b_k$ signed this from $k=1$ to $20$ ,The asymptotic of the titled series ...
2
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1answer
133 views

What can we know about “the half” of the generating series of Bessel function

I am interested in the series $$\sum_{n\geq 1}I_n(x)\lambda^n$$ which is not the full generating series of the modified Bessel function of the first kind because it starts from $n=1$ and not at $-\...
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1answer
172 views

What are the patterns of the sequence of polynomials? [closed]

In my research, I obtained a sequence of polynomials (I am only able to compute the first 4 of them): \begin{align} & f(2) = 1+t, \\ & f(3) = 1+4t+3t^2, \\ & f(4) = 1+6t+12t^2+7t^3, \\ &...
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8answers
2k views

Generating function in graph theory

I am looking for a simple illustration of generating functions in graph theory. So far, the matching polynomial seems to be the best. But I want something bit richer; at least a derivative should ...
4
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1answer
334 views

A generalization of binary Krawtchouk polynomials

I am looking for orthogonal polynomials $P_{d,m,n}$ that have their values at integers $i$ specified by the following generating function $(1-z)^i (1+z+z^2+ \ldots + z^d)^{n-i} = \sum_{m=0}^{i+d(n-i)}...
1
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0answers
196 views

Asymptotic behavior of coefficient of a complicated generating function

Given a generating function $$H(z)=\sum_{m=0}^\infty h_m z^m =\frac{P F(z)}{1-(1-P)F(z)}$$ where $0<P\leq 1$ and $$F(z)=1-\frac{\pi z\sqrt{c}}{6\boldsymbol{\mathrm{K}}(k')} $$ in which $\...
2
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0answers
146 views

Inverse of partial sums of general harmonic series

I would like to understand better the scaling of the following summation as a function of $r$ and $p > 1$: $$ S_r(p) := \sum_{m=1}^{r} \left( \sum_{k=r-m+1}^{r} \left( \frac{k^q}{\sum_{k'=1}^{r}...
10
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2answers
766 views

Proofs of some combinatorial identities

Just wondering if anyone knows any references in the literature to bijections corresponding to the following simple generating function identities. Let $B(z)=\dfrac{1}{\sqrt{1-4z}}$ and $C(z)=\dfrac{1-...
10
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1answer
711 views

Ramanujan's Lost Notebook page 1 first equation and OEIS sequence A260195

In the 1988 Narosa edition of Ramanujan's The Lost Notebook and Other Unpublished Papers, on the first line of page 1 is the following: $$ \Big(1+\frac1a\Big) \Bigg\{\frac{1}{(1-aq)(1-q/a)}+\frac{q(1+...
1
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0answers
112 views

Holonomic generating function

Let $\lambda$ denote a hook of size $d$ and $c(\Box)$ the content of $ \Box \in \lambda $. Let $ \text{Hooks}(d) $ be the set of hooks with $d$ boxes. Define \begin{align} B(d)&= \frac{1}{d!h^{d-1}...
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1answer
320 views

Infinite sum and product associated with the Weierstrass elliptic function [closed]

Can anyone help me figure out how the identity below was obtained? $ \frac{1}{\sqrt{(e_1-e_3)(e_2-e_3)}} = R \prod \limits_{n=1}^{\infty} \left(1 - \frac{1}{R^{4n}} \right)^{-4}\left(1 + \frac{1}{R^{...
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0answers
66 views

Construct generating functions of series of palindromic polynomials

I have a problem that is generating a series ($d=2,4,\ldots,20,\ldots$) of pairs of $4 d$-degree palindromic (self-reciprocal) polynomials. The first three members ($d=2,4,6$) of the first pair are: \...
1
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1answer
177 views

Asymptotics of Generating Functions

Given a generating function $A(x)$, are there any general techniques for finding the asymptotics of the associated sequence? For example, given the generating function satisfying $A(x) = 1 + x\cdot A\...
12
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0answers
352 views

What is this sequence?

This is again a question that I asked at Stack Exchange, but got no answer so far, so I am trying here. Let: $$ a_n=\sum_{k\ge0}(k+1) {n+2\brack k+2}(n+2)^kB_k$$ $B_k$ is the Bernoulli number. ${n\...
1
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0answers
92 views

Find bivariate generating function for two-dimensional sequence

How to find generating function for triangle of squares of elements in this sequence? I. e. for $1 + (1 + 4x)y + (1 + 9x + 16x^2)y^2 + ...$ ? It seems that ordinary approach with arithmetic ...
7
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0answers
92 views

Number of occurrences of certain generators in expressions in Coxeter groups

Let $W$ be a Coxeter group (finite or infinite) with (finite) set $S$ of Coxeter generators, and let $I \subseteq S$ be some subset. If $w\in W$ then I call $m_I(w)$ the minimum total number of ...
-2
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2answers
355 views

Expression for infinite product

can anyone show me how $$\displaystyle\frac{4}{R}\displaystyle\Pi_{n=1}^{\infty} \left(\frac{1+R^{-4n}}{1+R^{-4n+2}}\right)^4= \frac{1}{R}\left(1+2 \sum_{n=1}^ {\infty} \frac{1}{R^{2n(n+1)}}\...
7
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1answer
248 views

Counting some binary trees with lots of extra stucture

While working on some computations on Hilbert schemes, I came across the following combinatorial problem. Let $D(k,n)$ be the weighted number of binary trees (children are left/right) with $n$ ...
1
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3answers
121 views

asymptotics for coefficients of generating functions involving logarithms

I have a generating function that has a closed form like $1/(\log(z-a)+b)$ and I would like to get asymptotics for the size of the coefficients of it. I was going to use the methods in Chapter 5 of ...
3
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1answer
218 views

Multivariate Generating Function Related to Lambert $W$ Function and Counting Trees with a Certain Property

First, define a sequence $F_0,F_1,\dots$ of functions by $$F_0(x,z) = z,$$ $$F_k(x,z)=x\exp\left(F_{k-1}(x,z)\right) \quad \text{for }k\geq1.$$ So, for example, $$F_1(x,z) = x e^z, \quad F_2(x,z)=xe^{...
5
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2answers
252 views

Independent families of functions on $\omega$ of size continuum

In Hausdorff's article "Über zwei Sätze von G. Fichtenholz und L. Kantorovich''(1935) one can find the (simplified) proofs of the following two theorems: 1) There are continuum many essentially ...
4
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1answer
101 views

D-finiteness of Hilbert series of non-commutative invariant ring under reductive group

Let $G$ be reductive group over a field of characteristic $0$ ($GL_n$ fine for this question). Let $V$ be a linear representation of $G$. Then $G$ acts on the tensor algebra $T(V) = \bigoplus_{n \ge 0}...
2
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0answers
63 views

Is there an anti-commutator analog of Zassenhaus formula?

Is anyone familiar with an anti-commutator analog Zassenhaus formula? I have been able to find the anti-commutator analog of the BCH formula $$e^ABe^A= B + \{B,A\}+\frac{1}{2!}\{\{B,A\},A\}+ \frac{1}{...
5
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1answer
174 views

Reference request: Reduced reflection length in Coxeter groups

I recently read this paper, where the authors define on page 26 what they call the reduced reflection length. For that we take a Coxeter group $G$ with Coxeter generators $S$ and transpositions $T$. ...
2
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1answer
98 views

Recovering a distribution from sample averages?

I'm working on a problem where I have $n^2$ real numbers $x_{11},...,x_{nn}$, all drawn i.i.d. from the same distribution $F$. I don't observe each $x_{ij}$, but I do observe the $n$ means: $$\bar{x}...
7
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1answer
211 views

Bijective proof of formula for rooted binary forests

For $n\ge 1$, let $f(n)$ be the number of rooted complete (unordered) binary trees with $n$ leaves labeled from $1$ to $n$ ("complete binary" means that every vertex has either $0$ or $2$ children and ...
1
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1answer
207 views

partition theory: meet the COP

Recall that $(a;q)_0:=1,\,(a;q)_n=(1-a)(1-aq)(1-aq^2)\cdots(1-aq^{n-1})$ and $(a;q)_{\infty}=(1-a)(1-aq)(1-aq^2)\cdots$. Let's introduce the following (generalized) concept. A colored overpartition (...
2
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0answers
95 views

Unrestricting The Parameters of a Functional Equation

Good evening. I am looking into methods of generalization of Bernoulli polynomials. First, define $$\Phi_{N,k}(x)=\frac{1}{N}\sum_{j=0}^{N-1}\omega_N^{-jk}\exp\left(\omega_N^jx\right)$$ where $\...
3
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1answer
110 views

Recurrence relation asymptotics

A continuation from my two previous posts: I have got the following recurrence which describes polynomials: $$ C_n(a) = \sum\limits_{t=0}^{n-1} \binom{n-1}{t} a^{t(n-t)} C_t(a) $$ where $C_1(a)=C_0(...
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2answers
920 views

A mystery sequence

This question arose from the recent one, roots of a polynomial linked to mock theta function?. Let $$ g(x):=\sum_{k=0}^\infty x^k\prod_{j=1}^{k-1}(1 + x^j)^2\\=1+x+x^2+3 x^3+4 x^4+6 x^5+10 x^6+15 x^7+...
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0answers
176 views

A question about integer representation as a sum of two coprime integers

It is easy to see that every natural number $n$ can be written in a unique way $n = a+b$ where $gcd(a,b)=1$, $b>a$ and $b-a$ is minimal with this property. For instance if $n$ is odd the ...
8
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1answer
287 views

generalizing Wilf's conjecture: Uppuluri-Carpenter numbers

The complementary Bell numbers have the exponential generating function $$\sum_{n\geq0}\tilde{B}_nx^n=e^{1-e^x}.$$ Herb Wilf conjectured that $\tilde{B}_n=0$ only for $n=2$. By now, there are a few ...
5
votes
1answer
160 views

an algebra generated by some known series

Denote the e.g.f. for the number of (unordered) rooted labeled trees on $n$ nodes by $$\Phi(x)=\sum_{n\geq1}\frac{n^{n-1}}{n!}x^n.$$ And, the related series $\Psi(x)=\sum_{n\geq1}\frac{n^n}{n!}x^n$. ...
4
votes
1answer
183 views

Product of polynomial coefficients of a recurrence

A recurrence is given by $f[0]=2x$, $f[1]=3x^3-x^2+x+1$, $$ f[n]=(x^{2^n}+1)f[n-1]+(x^{2^n}+1)(x^{2^n-1}+1) $$ How does the PRODUCT of the nonzero coefficients of $f[n]$ scale with $n$?
4
votes
2answers
458 views

Total progeny of a Galton-Watson branching process - standard textbook question

While analyzing some parallel-computing related algorithm, I came across a probability distribution with a particularly nice property (at least to me), but I am unable to write it down explicitly. ...
0
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2answers
234 views

Generating function for products of complex Hermite polynomials

By making use of the generating function $$\sum_{m=0}^\infty \frac{H_m(x)}{m!} t^m=e^{-t^2 + 2xt} $$ for the real Hermite polynomials $H_m$, we get easily that $$(*)\quad \sum_{m,n=0}^\infty \frac{u^...