Some congruences on Delannoy numbers and Schröder numbers

2018 ◽  
Vol 14 (07) ◽  
pp. 2035-2041 ◽  
Author(s):  
Ji-Cai Liu ◽  
Long Li ◽  
Su-Dan Wang
Keyword(s):  
Schröder Numbers ◽  
Delannoy Numbers

The Delannoy numbers and Schröder numbers are given by [Formula: see text] respectively. We mainly prove that for any prime [Formula: see text], [Formula: see text] which was originally conjectured by Sun in 2011. A related congruence on the Delannoy numbers is also proved.

scholarly journals On Delannoy numbers and Schröder numbers

2011 ◽  
Vol 131 (12) ◽  
pp. 2387-2397 ◽  
Author(s):  
Zhi-Wei Sun
Keyword(s):  
Schröder Numbers ◽  
Delannoy Numbers

scholarly journals Hankel determinants of linear combinations of moments of orthogonal polynomials, II

2021 ◽  
Author(s):  
C. Krattenthaler
Keyword(s):  
Orthogonal Polynomials ◽  
Matrix Theory ◽  
Binomial Coefficients ◽  
Linear Recurrence ◽  
Hankel Determinants ◽  
Motzkin Numbers ◽  
Schröder Numbers ◽  
Delannoy Numbers ◽  
Dodgson Condensation ◽  
Full Proof

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.

A New Generalization of Delannoy Numbers

2020 ◽  
Vol 51 (4) ◽  
pp. 1729-1735
Author(s):  
Muhammet Cihat Dağli
Keyword(s):  
Delannoy Numbers

scholarly journals A determinantal expression and a recursive relation of the Delannoy numbers

2021 ◽  
Vol 13 (2) ◽  
pp. 442-449 ◽  
Author(s):  
Feng Qi
Keyword(s):  
Recursive Relation ◽  
Differentiable Functions ◽  
Delannoy Numbers ◽  
Derivatives Of ◽  
Determinantal Expression

Abstract In the paper, by a general and fundamental, but non-extensively circulated, formula for derivatives of a ratio of two differentiable functions and by a recursive relation of the Hessenberg determinant, the author finds a new determinantal expression and a new recursive relation of the Delannoy numbers. Consequently, the author derives a recursive relation for computing central Delannoy numbers in terms of related Delannoy numbers.

scholarly journals Alternative proofs of some formulas for two tridiagonal determinants

2018 ◽  
Vol 10 (2) ◽  
pp. 287-297 ◽  
Author(s):  
Feng Qi ◽  
Ai-Qi Liu
Keyword(s):  
Chebyshev Polynomials ◽  
Tridiagonal Matrix ◽  
Fibonacci Polynomials ◽  
Detailed Proof ◽  
Delannoy Numbers ◽  
Appl Math ◽  
Math Phys

Abstract In the paper, the authors provide five alternative proofs of two formulas for a tridiagonal determinant, supply a detailed proof of the inverse of the corresponding tridiagonal matrix, and provide a proof for a formula of another tridiagonal determinant. This is a companion of the paper [F. Qi, V. Čerňanová,and Y. S. Semenov, Some tridiagonal determinants related to central Delannoy numbers, the Chebyshev polynomials, and the Fibonacci polynomials, Politehn. Univ. Bucharest Sci. Bull. Ser. A Appl. Math. Phys. 81 (2019), in press.

Some properties of the Schröder numbers

2016 ◽  
Vol 47 (4) ◽  
pp. 717-732 ◽  
Author(s):  
Feng Qi ◽  
Xiao-Ting Shi ◽  
Bai-Ni Guo
Keyword(s):  
Schröder Numbers

ASYMPTOTICS OF THE WEIGHTED DELANNOY NUMBERS

2012 ◽  
Vol 08 (01) ◽  
pp. 175-188 ◽  
Author(s):  
ROB NOBLE
Keyword(s):  
Number Field ◽  
Asymptotic Expansions ◽  
Lattice Paths ◽  
Delannoy Numbers ◽  
Divisibility Properties ◽  
Special Case

The weighted Delannoy numbers give a weighted count of lattice paths starting at the origin and using only minimal east, north and northeast steps. Full asymptotic expansions exist for various diagonals of the weighted Delannoy numbers. In the particular case of the central weighted Delannoy numbers, certain weights give rise to asymptotic coefficients that lie in a number field. In this paper we apply a generalization of a method of Stoll and Haible to obtain divisibility properties for the asymptotic coefficients in this case. We also provide a similar result for a special case of the diagonal with slope 2.

Sign in / Sign up

Export Citation Format

Share Document