Related Number Sequences: The Motzkin Numbers, The Fine Numbers, and the Schröder Numbers

AbstractWe present a formula that expresses the Hankel determinants of a linear combination of length $$d+1$$ d + 1 of moments of orthogonal polynomials in terms of a $$d\times d$$ d × d determinant of the orthogonal polynomials. This formula exists somehow hidden in the folklore of the theory of orthogonal polynomials but deserves to be better known, and be presented correctly and with full proof. We present four fundamentally different proofs, one that uses classical formulae from the theory of orthogonal polynomials, one that uses a vanishing argument and is due to Elouafi (J Math Anal Appl 431:1253–1274, 2015) (but given in an incomplete form there), one that is inspired by random matrix theory and is due to Brézin and Hikami (Commun Math Phys 214:111–135, 2000), and one that uses (Dodgson) condensation. We give two applications of the formula. In the first application, we explain how to compute such Hankel determinants in a singular case. The second application concerns the linear recurrence of such Hankel determinants for a certain class of moments that covers numerous classical combinatorial sequences, including Catalan numbers, Motzkin numbers, central binomial coefficients, central trinomial coefficients, central Delannoy numbers, Schröder numbers, Riordan numbers, and Fine numbers.

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Pascal's Prism

The Mathematical Gazette ◽

10.1017/s0025557200004447 ◽

2012 ◽

Vol 96 (536) ◽

pp. 213-220

Author(s):

Harlan J. Brothers

Keyword(s):

Probability Theory ◽

Fractal Geometry ◽

Euclidean Geometry ◽

Fibonacci Series ◽

Pascal’S Triangle ◽

Number Sequences ◽

Definition Of ◽

Image Position ◽

Pascal's Triangle

Pascal's triangle is well known for its numerous connections to probability theory [1], combinatorics, Euclidean geometry, fractal geometry, and many number sequences including the Fibonacci series [2,3,4]. It also has a deep connection to the base of natural logarithms, e [5]. This link to e can be used as a springboard for generating a family of related triangles that together create a rich combinatoric object.2. From Pascal to LeibnizIn Brothers [5], the author shows that the growth of Pascal's triangle is related to the limit definition of e.Specifically, we define the sequence sn; as follows [6]:

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Diaphony, a measure of uniform distribution, and the Patterson function

Acta Crystallographica Section A Foundations and Advances ◽

10.1107/s2053273315007123 ◽

2015 ◽

Vol 71 (4) ◽

pp. 382-391 ◽

Cited By ~ 2

Author(s):

Wolfgang Hornfeck ◽

Philipp Kuhn

Keyword(s):

Uniform Distribution ◽

Structure Factor ◽

Interplanar Spacing ◽

Theory Approach ◽

Point Sets ◽

Patterson Function ◽

Theoretic Concept ◽

Number Sequences ◽

Fourier Theory ◽

Factor Equation

This paper reviews the number-theoretic concept ofdiaphony, a measure of uniform distribution for number sequences and point sets based on a Fourier theory approach, and its relation to crystallographic concepts like the largest interplanar spacing of a lattice, the structure-factor equation and the Patterson function.

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p-adic numbers encode complex networks

Scientific Reports ◽

10.1038/s41598-020-79507-4 ◽

2021 ◽

Vol 11 (1) ◽

Author(s):

Hao Hua ◽

Ludger Hovestadt

Keyword(s):

Complex Networks ◽

Random Graph ◽

Genetic Interaction ◽

Hierarchical Structures ◽

Number System ◽

Hierarchical Organization ◽

Natural Representation ◽

Multimodal Distributions ◽

Number Sequences ◽

Er Model

AbstractThe Erdős-Rényi (ER) random graph G(n, p) analytically characterizes the behaviors in complex networks. However, attempts to fit real-world observations need more sophisticated structures (e.g., multilayer networks), rules (e.g., Achlioptas processes), and projections onto geometric, social, or geographic spaces. The p-adic number system offers a natural representation of hierarchical organization of complex networks. The p-adic random graph interprets n as the cardinality of a set of p-adic numbers. Constructing a vast space of hierarchical structures is equivalent for combining number sequences. Although the giant component is vital in dynamic evolution of networks, the structure of multiple big components is also essential. Fitting the sizes of the few largest components to empirical data was rarely demonstrated. The p-adic ultrametric enables the ER model to simulate multiple big components from the observations of genetic interaction networks, social networks, and epidemics. Community structures lead to multimodal distributions of the big component sizes in networks, which have important implications in intervention of spreading processes.

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Development of a direct NGM(1,1) prediction model based on interval grey numbers

Grey Systems Theory and Application ◽

10.1108/gs-07-2020-0097 ◽

2021 ◽

Vol ahead-of-print (ahead-of-print) ◽

Author(s):

Ye Li ◽

Yuanping Ding ◽

Yaqian Jing ◽

Sandang Guo

Keyword(s):

Genetic Algorithm ◽

Markov Model ◽

Prediction Model ◽

Real Number ◽

Content Type ◽

Optimal Weight ◽

Practical Applications ◽

Number Sequences ◽

Integral Development ◽

Grey Number

PurposeThe purpose of this paper is to construct an interval grey number NGM(1,1) direct prediction model (abbreviated as IGNGM(1,1)), which need not transform interval grey numbers sequences into real number sequences, and the Markov model is used to optimize residual sequences of IGNGM(1,1) model.Design/methodology/approachA definition equation of IGNGM(1,1) model is proposed in this paper, and its time response function is solved by recursive iteration method. Next, the optimal weight of development coefficients of two boundaries is obtained by genetic algorithm, which is designed by minimizing the average relative error based on time weighted. In addition to that, the Markov model is used to modify residual sequences.FindingsThe interval grey numbers’ sequences can be predicted directly by IGNGM(1,1) model and its residual sequences can be amended by Markov model. A case study shows that the proposed model has higher accuracy in prediction.Practical implicationsUncertainty and volatility information is widespread in practical applications, and the information can be characterized by interval grey numbers. In this paper, an interval grey numbers direct prediction model is proposed, which provides a method for predicting the uncertainty information in the real world.Originality/valueThe main contribution of this paper is to propose an IGNGM(1,1) model which can realize interval grey numbers prediction without transforming them into real number and solve the optimal weight of integral development coefficient by genetic algorithm so as to avoid the distortion of prediction results. Moreover, the Markov model is used to modify residual sequences to further improve the modeling accuracy.

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