Kunle Adegoke, Robert Frontczak, Taras Goy
We study certain series with Catalan numbers and reciprocal Catalan numbers, respectively, and provide seemingly new closed form evaluations of these series with Fibonacci (Lucas) entries. In addition, we state some combinatorial sums that can be inferred from the series.
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We use the inversion of coefficient arrays to define dual polynomials to the Fibonacci and Catalan-Fibonacci polynomials, and we explore the properties of these new polynomials sequences. Many of the arrays involved are Riordan arrays. Direct links to the counting of Motzkin paths by different statistics emerge.
Solutions for $k$-generalized Fibonacci numbers using Fuss-Catalan numbers
该研究提出k-广义斐波那契数Fₖ(n)的新表达式，利用Fuss-Catalan数给出其特征多项式根的显式形式，量化渐近近似精度，提供任意初始条件下的递推基，扩展前人结果并发现多变量Dickson多项式关联及伴随矩阵新恒等式。
Identities and Generating Functions of Products of Generalized Fibonacci numbers, Catalan and Harmonic Numbers
本文研究了广义 Fibonacci 与 Lucas 数类的性质，获得了著名 Fibonacci 恒等式的类比及乘积卷积新恒等式，并写下了其乘积、多截、调和数与 Catalan 数的生成函数。
From Fibonacci to Catalan permutations
众所周知，避免任何3-长度模式的排列都是用Catalan数枚举的。如果同时避免123、132和213这三个模式，我们得到一类由斐波那契数枚举的排列。我们从这些排列开始，通过适当的“概括”它们，使一两个禁止的模式消失。在这样一种方式中，我们发现几类排列列举的整数序列之间的斐波那契和加泰罗尼亚数字。对于每个类，我们根据排列的长度提供生成函数。此外，作为一个结果，我们引入了一种“连续性”的号码序列枚举这些类的排列。
This (partly expository) paper originated from the study of Hankel determinants of convolution powers of Catalan numbers and of Narayana polynomials. This led to some Hankel determinants of signed Catalan numbers whose values are multiples of Fibonacci numbers and to some Hankel determinants of signed central binomial coefficients whose values are multiples of Lucas numbers. Most proofs are computational but we also include a combinatorial one due to Christian Krattenthaler. Finally we formulate some conjectures.
We study a certain family of infinite series with reciprocal Catalan numbers. We first evaluate two special candidates of the family in closed form, where we also present some Catalan-Fibonacci relations. Then we focus on the general properties of the family and prove explicit formulas, including two types of integral representations.
In this note, we present some basic properties of $q$-Fibonacci numbers and their relationship to the $q$-golden ratio and Catalan numbers. We then use this relationship to give a short proof of a combinatorial identity.
The notion of $(a,b)$-cores is closely related to rational $(a,b)$ Dyck paths due to Anderson's bijection, and thus the number of $(a,a+1)$-cores is given by the Catalan number $C_a$. Recent research shows that $(a,a+1)$ cores with distinct parts are enumerated by another important sequence- Fibonacci numbers $F_a$. In this paper, we consider the abacus description of $(a,b)$-cores to introduce the natural grading and generalize this result to $(a,as+1)$-cores. We also use the bijection with Dyck paths to count the number of $(2k-1,2k+1)$-cores with distinct parts. We give a second grading to …
The \emph{Hosoya triangle} is a triangular array where every entry is a product of two Fibonacci numbers. We use the geometry of this triangle to find new identities related to Fibonacci numbers. We give geometric interpretation for some well-known identities of Fibonacci numbers. For instance, the Cassini identity and the Catalan identity. We also extend some identities that hold in the Pascal triangle to the Hosoya triangle. For instance, the hockey stick extends from binomials to products of Fibonacci numbers and the rhombus property extends a binomial identity from the Pascal triangle to a…
Following Benjamin et al., a matrix with entries being sums of two neighbouring Catalan numbers is considered. Its LU-decomposition is given, by guessing the results and later prove it by computer algebra, with lots of human help. Specializing a parameter, the determinant turns out to be a Fibonacci number with odd index, confirming earlier results, obtained back then by combinatorial methods.
Results of this paper concern $n$-determinants which we defined in the paper \cite{jan}. In the paper \cite{jabo}, $2$-determinants are considered. In this paper, we extend results from \cite{jabo} on $n$-determinants by proving that Cassini, d'Ocagne, Catalan and Vajda identities may be extended to hold for $n$-step Fibonacci numbers.
We calculate moments of the so-called Kesten distribution by means of the expansion of the denominator of the density of this distribution and then integrate all summands with respect to the semicircle distribution. By comparing this expression with the formulae for the moments of Kesten's distribution obtained by other means, we find identities involving polynomials whose power coefficients are closely related to Catalan numbers, Catalan triangles, binomial coefficients. Finally, as applications of these identities we obtain various interesting relations between the aforementioned numbers, al…
On Series Involving Cubed Catalan Numbers
本文利用广义二项式系数恒等式和Dougall的结果，推导出涉及卡特兰数立方的级数家族，并建立包含四次方卡特兰数的级数家族，还推广了Bauer级数用于1/π，并得到一些Ramanujan-like级数用于1/π²和1/π³。
In this paper, we present a formula for generating various exotic series in the spirit of Ovidiu Furdui and Alina Sintamarian. Our new series (evaluated in closed form) involve Bernoulli, harmonic, and Catalan numbers. Also Stirling numbers of the second kind, other special numbers, and exponential polynomials. The results include series identities with Laguerre polynomials and derangement polynomials.
Cassini-Catalan Determinants via Ramanujan's Theta Identity
本文利用拉马努金 theta 恒等式统一推导 Fibonacci 数的 Cassini 与 Catalan 行列式，并推广到划分细化版本及高阶递推，揭示其模形式来源与组合意义。
A combinatorial structure, $\mathcal{F}$, with counting sequence $\{a_n\}_{n\ge 0}$ and ordinary generating function $G_\mathcal{F}=\sum_{n\ge0} a_n x^n$, is positive algebraic if $G_\mathcal{F}$ satisfies a polynomial equation $G_\mathcal{F}=\sum_{k=0}^N p_k(x)\,G_\mathcal{F}^k $ and $p_k(x)$ is a polynomial in $x$ with non-negative integer coefficients. We show that every such family is associated with a normed $\mathbf{n}$-magma. An $\mathbf{n}$-magma with $\mathbf{n}=(n_1,\dots, n_k)$ is a pair $\mathcal{M}$ and $\mathcal{F}$ where $\mathcal{M}$ is a set of combinatorial structures and $\m…
The moments of the Lucas polynomials and of the Chebyshev polynomials of the first kind are (multiples of) central binomial coefficients and the moments of the Fibonacci polynomials and of the Chebyshev polynomials of the second kind are Catalan numbers. In this survey paper we present some generalizations of these results together with various q-analogues.
Fibonacci Polyominoes: Refined Enumeration and Dyck-Path Bijections
本文研究 Fibonacci 多联骨牌的按面积和半周长枚举，通过生成函数得到含 Catalan 数的精细公式，并建立与带标签 Dyck 路径的双射，将统计量转化为峰与返回，还研究等高连续列并得到 Motzkin 数控制的精细化。
We use probability urn models to discover some known and unknown series identities involving Fibonacci numbers.
Some power series extensions of Fibonacci and Lucas polynomials
本研究聚焦形式幂级数领域，探讨可作为偶/奇下标斐波那契、卢卡斯多项式插值的形式幂级数，拓展了相关多项式的幂级数研究方向。
Generalization of an Identity of Andrews
我们考虑一个由G. G发现的将斐波那契数与帕斯卡三角形联系起来的恒等式。E.安德鲁斯几位作者提供了证明这一身份，他们都相当参与或依赖于复杂的数论参数。不仅给出了一个简单初等的证明，而且证明了该恒等式推广到了Pascal三角形以外的数组。作为应用，我们得到的身份三项式系数和加泰罗尼亚的三角形斐波那契数。
Let $β=\frac{1+\sqrt{5}}{2}$, $(a_n)_{n \in \mathbb{N}^+}$ be a non-uniform morphic sequence involving the infinite Fibonacci word and $(δ(n))_{n \in \mathbb{N}^+}$ be a positive sequence such that for all positive integers $n$, $δ(n)=\frac{1}{\sqrt{5}}\sum_{j \geq 0}ε_jβ^{j+2}$ if the unique Zeckendorf expansion of $n$ is $n=\sum_{j \geq 0}ε_jF_{j+2}$ with Fibonacci numbers $F_0,F_1,F_2...$. We define and study some Dirichlet series in the form of $\sum_{n\geq 1}\frac{a_n}{(δ(n))^s}$ and relations between them. Moreover, we compute the values of some infinite series involving the infinite Fib…
Series Associated with Harmonic Numbers, Fibonacci Numbers and Central Binomial Coefficients $\binom{2n}{n}$
本文通过直接变换而非传统超几何方法，发现了涉及中心二项式系数、调和数与斐波那契数的多种级数，并给出了 $\zeta(2)$ 的新级数表示。
We mainly show a supercongruence for a truncated series with cubes of Catalan numbers which extends a result by Zhi-Wei Sun.
The coefficients of a Fibonacci power series
我们给出一个简短的证明，证明了级数（1-x）（1-x^2）（1-x^3）（1-x^5）（1-x^8）（1-x^13）（1-x^21）.等于-1，0或1，并且它们中的大多数等于0。
Touchard's identity and a detailed determination of the radius of convergence of the Catalan series
本文通过触德递推关系证明了卡特兰级数收敛半径为1/4，并结合经典下界得出精确结果。
We confirm a conjectural supercongruence involving Catalan numbers, which is one of the 100 selected open conjectures on congruences of Sun. The proof makes use of hypergeometric series identities and symbolic summation method.
Exercises for A Hyper-Catalan Series Solution to Polynomial Equations, and the Geode
本文为“多项式方程的超加泰罗尼亚级数解与测地线”一文配套编写习题并给出解答，以帮助读者理解和练习相关内容。
We evaluate some types of infinite series with balancing and Lucas-balancing polynomials in closed form. These evaluations will lead to some new curious series for $π$ involving Fibonacci and Lucas numbers. Our findings complement those of Castellanos from 1986 and 1989.
Harmonic Catalan Number Series via Half-Integer Binomial Coefficients
该研究通过半整数参数的交替二项式恒等式，结合广义二项式系数的显式表示与相关恒等式，推导出含卡特兰数、调和数的闭式求和公式，为转化恒等式提供了系统机制。
I study Hankel determinants of a class of sequences which can be interpreted as generalizations of the Catalan numbers and the central binomial coefficients. They follow a modular pattern with a frequent appearance of zeroes, so that the theory of orthogonal polynomials is not applicable. Even so our theorems and conjectures show some similarity with the relations between Catalan numbers and Fibonacci polynomials or between central binomial coefficients and Lucas polynomials.
Pattern avoidance in "flattened" partitions
使一个集合划分变平（向Mathematica道歉）意味着通过擦除块之间的分隔符来形成一个排列。当然，结果取决于如何列出块。对于通常的列表--每个块中的条目不断增加，并且块按照其第一个条目的递增顺序排列--我们计算[n]的分区，这些分区的扁平化避免了单个3个字母的模式。出现五个计数序列：零序列，2的幂，斐波那契数，卡塔兰数，卡塔兰数的二项式变换。
We show that q-Catalan numbers, q- central binomial coefficients and q- Narayana polynomials are moments of q-analogues of Fibonacci and Lucas polynomials and related polynomials.
Using the notion of series parallel interval order, we propose a unified setting to describe Dyck lattices and Tamari lattices (two well known lattice structures on Catalan objects) in terms of basic notions of the theory of posets. As a consequence of our approach, we find an extremely simple proof of the fact that the Dyck order is a refinement of the Tamari one. Moreover, we provide a description of both the weak and the strong Bruhat order on 312-avoiding permutations, by recovering the proof of the fact that they are isomorphic to the Tamari and the Dyck order, respectively; our proof, wh…
The Fibonacci number is the residue of a rational function, from which follows that Fibonacci number summation identities can be derived with the integral representation method, a method also used to derive combinatorial identities. A number of weighted generalized Fibonacci number summation identities are derived this way. From these identities some infinite series, generating functions and convolution identities are obtained. In addition, some weighted generalized Fibonacci number summation identities with binomial coefficients are derived. Many examples of both types of summation identities…
Two new identities about Catalan numbers are treated with Zeilberger's algorithm and Watson's hypergeometric series evaluation.
On the coefficients of a Fibonacci power series
本文给出了形式幂级数（1-x）（1-x^2）（1-x^3）（1-x^5）（1-x^8）（1-x^13）.特别地，我们证明了所有的系数都等于-1，0或1。
Finite Interpretation of the Hyper-Catalan Series Zero and its Powers
本文将单变量几何多项式形式级数零的超卡塔兰数生成级数解释为各层级上的有限恒等式，并探讨了其幂$\mathbf{S}^r$系数的组合推导。
Nested Hilbert schemes and the nested q,t-Catalan series
本文研究了平面上点的光滑套Hilbert格式$ Hil{n,n-1}$的切空间，给出了计算$\Hil{n,n-1}$上$ Hil{n,n-1}$-等变局部自由层的欧拉示性数的一般公式.应用我们的结果到一个特定的层，我们推测，该结果是一个多项式的变量$q$和$t$与非负整数系数。我们称这个正多项式为\textsl{the nested $q,t$-Cat alan series}，因为它具有许多类似于$q,t $ -Catalan级数的性质.
We give a simple recursion labeled by binary sequences which computes rational $q,t$-Catalan power series, both in relatively prime and non relatively prime cases. It is inspired by, but not identical to recursions due to B. Elias, M. Hogancamp, and A. Mellit, obtained in their study of link homology. We also compare our recursion with the Hogancamp-Mellit's recursion and verify a connection between the Khovanov-Rozansky homology of $N,M$-torus links and the rational $q,t$-Catalan power series for general positive $N,M.$
Chebyshev polynomials, their remarkable properties and connection with Catalan numbers
该研究证明经重整化的相邻切比雪夫多项式之比的形式幂级数可逼近卡特兰数生成函数，用连分数完成证明并回顾相关基础性质。
This paper presents a number of identities for Dirichlet series and series with Stirling numbers of the first kind. As coefficients for the Dirichlet series we use Cauchy numbers of the first and second kinds, hyperharmonic numbers, derangement numbers, binomial coefficients, central binomial coefficients, and Catalan numbers.
As is well-known, the ratio of adjacent Fibonacci numbers tends to phi = (1 + sqrt(5))/2, and the ratio of adjacent Tribonacci numbers (where each term is the sum of the three preceding numbers) tends to the real root eta of X^3 - X^2 - X - 1 = 0. Letting alpha(n) denote the corresponding ratio for the generalized Fibonacci numbers, where each term is the sum of the n preceding, we obtain rapidly converging series for alpha(n), 1/alpha(n), and 1/(2-alpha(n)).
The Lucas sequence is a sequence of polynomials in s, and t defined recursively by {0}=0, {1}=1, and {n}=s{n-1}+t{n-2} for n >= 2. On specialization of s and t one can recover the Fibonacci numbers, the nonnegative integers, and the q-integers [n]_q. Given a quantity which is expressed in terms of products and quotients of nonnegative integers, one obtains a Lucas analogue by replacing each factor of n in the expression with {n}. It is then natural to ask if the resulting rational function is actually a polynomial in s and t with nonnegative integer coefficients and, if so, what it counts. The…
We study a family of finite--dimensional representations of the hyperspecial parabolic subalgebra of the twisted affine Lie algebra of type $\tt A_2^{(2)}$. We prove that these modules admit a decreasing filtration whose sections are isomorphic to stable Demazure modules in an integrable highest weight module of sufficiently large level. In particular, we show that any stable level $m'$ Demazure module admits a filtration by level $m$ Demazure modules for all $m\ge m'$. We define the graded and weighted generating functions which encode the multiplicity of a given Demazure module and establish…
q,t-Fuss-Catalan numbers for complex reflection groups
在类型A中，q，t-Fuss -Catalan数可以定义为与对称群S_n有关的模的双阶化Hilbert级数.我们将这种构造推广到（有限）复反射群，并展示了这些多项式在q和t中的一些很好的代数和组合性质。最后，我们提出了一个想法，这些多项式可以与一些分次希尔伯特系列的模块中产生的理性切雷德尼克代数的背景下。这是正在进行的工作。
A finitization of the Catalan numbers $ C_n $ can be defined as Euler characteristics of an algebraic structure. We conjecture the existence of a $ q $-deformed version of such structure, and provide evidence for the first two non-trivial cases. This $ q $-deformed version will likely categorify series arise from identities of the Roger-Ramanujan type.
Let $\{P_n\}_{n\geq 0}$ denote the Catalan-Larcombe-French sequence, which naturally came up from the series expansion of the complete elliptic integral of the first kind. In this paper, we prove the strict log-concavity of the sequence $\{\sqrt[n]{P_n}\}_{n\geq 1}$, which was originally conjectured by Sun. We also obtain the strict log-concavity of the sequence $\{\sqrt[n]{V_n}\}_{n\geq 1}$, where $\{V_n\}_{n\geq 0}$ is the Fennessey-Larcombe-French sequence arising in the series expansion of the complete elliptic integral of the second kind.
Asymptotics for partitions over the Fibonacci numbers and related sequences
本文结合经典分拆理论与 Fibonacci Dirichlet 级数的新结果，首次给出整数表示为非互异斐波那契数之和的方式数 $p_F(n)$ 的渐近公式，并将类似结果推广到一般线性递推序列。
Combinatorial connections in snake graphs: Tilings, lattice paths, and perfect matchings
本文引入三角蛇图，建立其路径、底层蛇图完美匹配与铺砌之间的双射，并据此给出直蛇图完美匹配数的 Hankel 行列式表达及 Fibonacci、Pell 序列的行列式来源。
Time-evolution of the Rule 150 cellular automaton activity from a Fibonacci iteration
单种子细胞规则150自动机的总活动不像其他基本细胞自动机那样遵循一步迭代，而是可以作为两步向量或串迭代来求解，这可以被视为从长度增加的向量序列生成时间序列的斐波那契迭代的推广。这允许比通过模拟整个时空过程或者甚至通过使用闭合表达式更有效地计算总活动时间序列。
A Filtration of (q,t)-Catalan numbers
我们定义了新的推广（q，t）-Catalan数应用nabla算子的列分区索引的k-Schur函数。在一些特殊的情况下，我们给出了一个组合的解释，这些号码使用配置的Dyck路径。在某些特殊情况下，我们也将这些新多项式解释为用微分算子构造的交替对角调和函数的显式子模的Hilbert级数。
In this note, we explore certain determinantal descriptions of the Robbins numbers. Techniques used for this include continued fractions, Riordan arrays and series inversion. Proven and conjectured representations involve the determinants of both Hankel and symmetric matrices. In specific cases, links are drawn to centered polygonal numbers, and to heptagons and nonagons. We conjecture a Hankel transform determinant for the Robbins numbers related to the Fibonacci and the Catalan numbers.
In type A, the q,t-Fuss-Catalan numbers can be defined as a bigraded Hilbert series of a module associated to the symmetric group. We generalize this construction to (finite) complex reflection groups and, based on computer experiments, we exhibit several conjectured algebraic and combinatorial properties of these polynomials with non-negative integer coefficients. We prove the conjectures for the dihedral groups and for the cyclic groups. Finally, we present several ideas how the q,t-Fuss-Catalan numbers could be related to some graded Hilbert series of modules arising in the context of ratio…
Multiparameter Fuss--Catalan numbers with application to algebraic equations
本文系统整理了多参数Fuss-Catalan数的定义、恒等式及历史，并提出了代数方程无穷级数解绝对收敛域的新充要条件。
Convoluted convolved Fibonacci numbers
卷积斐波那契数F_j^（r）定义为（1-z-z^2）^{-r}= \sum_{j>=0} F_{j+1} ^（r）z^j.本文考虑了一些可用卷积斐波那契数表示的相关数.这些数字出现在某个数字理论常数的数值计算中。本文是对变换{1/n} \sum_{d|n} mu（d）f（z^d）^{n/d}的一个案例研究，其中f是任何形式级数，mu是Moebius函数），在题为“形式级数Witt变换”的配套论文中对此进行了研究。
Schreier Sets of Intervals, Super-Schreier Sets, and Catalan Numbers
该数学研究定义区间型施赖尔集、$k$-超级施赖尔集，证明其计数序列分别满足特定特征多项式对应的递推关系与带多项式余项的斐波那契型递推。
We consider the Cauchy problem for the Volterra chain with an initial condition equal to 0 in one node and 1 in the others. It is shown that this problem admits an exact solution in terms of the Bessel functions. The Taylor series arising here are related to the exponential generating function for Catalan numbers.
Combinatorial study of the q-Catalan triangle and its generalizations
本文通过七种组合结构探讨q-Catalan三角形的组合意义，引入镜像多项式并证明其双重递推关系，提出q,p-Catalan三角形及多变量推广，首次完整证明基于312-avoiding排列的q-Catalan三角形定理。
Pattern avoiding machines were introduced recently by Claesson, Cerbai and Ferrari as a particular case of the two-stacks in series sorting device. They consist of two restricted stacks in series, ruled by a right-greedy procedure and the stacks avoid some specified patterns. Some of the obtained results have been further generalized to Cayley permutations by Cerbai, specialized to particular patterns by Defant and Zheng, or considered in the context of functions over the symmetric group by Berlow. In this work we study pattern avoiding machines where the first stack avoids a pair of patterns …
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