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Simple closed form Hankel transforms based on the central coefficients of certain Pascal-like triangles
[article]

2006
*
arXiv
*
pre-print

We study the Hankel transforms

arXiv:math/0605169v1
fatcat:szik4tm37ndexoehhe73qowsti
*of*sequences related to the central coefficients*of**a**family**of**Pascal*-*like**triangles*. ... The mechanism*of*Riordan arrays is used to elucidate the structure*of*these transforms. ... Introduction This note concerns the characterization*of*the Hankel transfoms*of*the central coefficients T (2n, n, r)*of**a**family**of**Pascal*-*like**triangles*that are parameterised by an integer r. ...##
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The γ-Vectors of Pascal-like Triangles Defined by Riordan Arrays
[article]

2018
*
arXiv
*
pre-print

We are led to the γ-matrices

arXiv:1804.05027v1
fatcat:ogzkbkdt2zbepklvpxst67bsiy
*of**a*one-parameter*family**of**generalized*Narayana*triangles*. Thus these matrices*generalize*the matrix*of*γ-vectors*of*the associahedron. ... We define and characterize the γ-matrix*associated*to*Pascal*-*like*matrices that are defined by ordinary and exponential Riordan arrays. ... Conclusion It is the case that the set*of**Pascal*-*like*matrices defined by Riordan arrays is*a*restricted one. ...##
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Counting odd numbers in truncations of Pascal's triangle
[article]

2018
*
arXiv
*
pre-print

Presented here is

arXiv:1807.08181v2
fatcat:xuazxff5ijfd3ko6b7wqz7ehgm
*a**family**of*natural truncations*of*Pascal's*triangle*that*generalize**a*kind*of**Catalan**triangle*. ...*A*"truncation"*of*Pascal's*triangle*is*a*triangular array*of**numbers*that satisfies the usual*Pascal*recurrence but*with**a*boundary condition that declares some terminal set*of**numbers*along each row*of*... For the rest*of*the paper, t denotes*a*fixed positive integer, which we think*of*informally as designating the first truncated row*of*the*associated**Pascal*-*like*integer array. ...##
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A note on a one-parameter family of non-symmetric number triangles

2012
*
Opuscula Mathematica
*

This concerns, in particular,

doi:10.7494/opmath.2012.32.4.661
fatcat:444jmg7xtbd3tgfb326f4sggvi
*a**generalized*Appell sequence*of*homogeneous polynomials whose coefficient set can be treated as*a*one-parameter*family**of*non-symmetric*triangles**of*fractions. ... The discussion*of*its properties, similar to those*of*the ordinary*Pascal**triangle*(which itself does not belong to the*family*), is carried out in this paper. ... within project PEst-C/MAT/ UI4106/2011*with*COMPETE*number*FCOMP-01-0124-FEDER-022690. ...##
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The second part of on duality triads' paper
[article]

2004
*
arXiv
*
pre-print

In this paper Fibonomial

arXiv:math/0402288v1
fatcat:74trw42gvnew7aizrtzi6idjpa
*triangle*and further*Pascal*-*like**triangles*including q-Gaussian one are given explicit interpretation as discrete time dynamical systems as it is the case*with*all duality triads ... Notation, enumeration*of*formulas and references is therefore to be continued hereby. ...*Catalan*triad and*triangle*Let C n,k denotes the*number**of*pairs*of*nonintersecting paths*of*length n and distance k as defined in [36] . ...##
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Generalized Eulerian Triangles and Some Special Production Matrices
[article]

2018
*
arXiv
*
pre-print

Using the previously defined T transform, we

arXiv:1803.10297v1
fatcat:2qr7xnz625d4bkst2g327v54yi
*associate*these*generalized*Eulerian*triangles*to*triangles*defined by*Catalan**generating*functions. ... We show how some special production matrices may be used to define*families**of**generalized*Eulerian*triangles*. ... The*generating*function*of*the*Catalan**numbers*C n = 1 n+1 2n n will be denoted by c(x). We have c(x) = 1 − √ 1 − 4x 2x . ...##
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Function Series, Catalan Numbers, and Random Walks on Trees

2005
*
The American mathematical monthly
*

We also thank Glenn Easley for his help

doi:10.2307/30037599
fatcat:xjess7f4u5cjbccjnxqre626oe
*with*the graphics. ... As in the case*of*the*Catalan**numbers*, the*generalized**Catalan**numbers*can be derived by means*of**a**Pascal*-*like**triangle*as follows. Fix an integer k bigger than 1. ... Graph*associated**with*the*generalized**Catalan**numbers**a*n,3 . ...##
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Function Series, Catalan Numbers, and Random Walks on Trees

2005
*
The American mathematical monthly
*

We also thank Glenn Easley for his help

doi:10.1080/00029890.2005.11920251
fatcat:t7td45feufaexgsi7jcg2ia6cq
*with*the graphics. ... As in the case*of*the*Catalan**numbers*, the*generalized**Catalan**numbers*can be derived by means*of**a**Pascal*-*like**triangle*as follows. Fix an integer k bigger than 1. ... Graph*associated**with*the*generalized**Catalan**numbers**a*n,3 . ...##
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Generalized Pascal Triangles and Toeplitz Matrices
[article]

2009
*
arXiv
*
pre-print

This equality allows us to evaluate

arXiv:0901.2597v1
fatcat:u7wbe3xajvd2rbwdwuecn6g7ou
*a*few determinants*of**generalized**Pascal*matrices*associated*to certain sequences. ... The purpose*of*this article is to study determinants*of*matrices which are known as*generalized**Pascal**triangles*(see [1]). ... The first author would*like*to thank IPM for the financial support. ...##
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Pascal arrays: counting Catalan sets
[article]

2006
*
arXiv
*
pre-print

Certain

arXiv:math/0612572v1
fatcat:hxs6u6vlqrbn5fdmfehljrffja
*of*these coincide*with*well known (but heretofore isolated) structures. The remainder are new. ... Motivated by representation theory we exhibit an interior structure to*Catalan*sequences and many generalisations thereof. ... Note that the array*of*cardinalities for this*Pascal*array is not the*Catalan**triangle*but the ordinary*Pascal**triangle*. (This example is the reason for our use*of*the term in*general*.) ...##
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On a Central Transform of Integer Sequences
[article]

2020
*
arXiv
*
pre-print

Starting from simple sequences

arXiv:2004.04577v1
fatcat:kyyk34m3urdjti3khjrgdu5uxq
*with*elementary rational*generating*functions, we obtain many sequences*of*combinatorial significance. ... We use the concept*of*the half*of**a*lower-triangular matrix to define*a*transformation on integer sequences. ... The C transform*of*the sequence*with**generating*function 1+ax 1−x 2 is the INVERT(*a*− 1) transform*of*the*Catalan**numbers*. Proof. ...##
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Generalized Pascal triangles and Toeplitz matrices

2009
*
The Electronic Journal of Linear Algebra
*

This equality allows for the evaluation

doi:10.13001/1081-3810.1331
fatcat:wh55um6n7bh5fpgic2udmukqzu
*of**a*few determinants*of**generalized**Pascal*matrices*associated**with*certain sequences. ... The purpose*of*this article is to study determinants*of*matrices which are known as*generalized**Pascal**triangles*(see R. Bacher. Determinants*of*matrices related to the*Pascal**triangle*. J. Théor. ... We would*like*to thank the referee for the careful reading*of*an earlier version*of*this article and for suggestions and comments. The first author would*like*to thank IPM for their financial support. ...##
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A LINK BETWEEN ORDERED TREES AND GREEN-RED TREES

2016
*
Journal of the Korean Mathematical Society
*

The r-ary

doi:10.4134/jkms.2016.53.1.187
fatcat:ejygx7cnejdtvlv775ryukaoke
*number*sequences given by ... We note that ∆ 1 = (1/(1 − z), z/(1−z)) and ∆ 2 = (C, zC) are the*Pascal**triangle*and the*Catalan**triangle*, respectively. ... One*of*the important*generalizations**of*the*Catalan**numbers*are the r-ary*numbers*b (r) n = 1 (r−1)n+1 rn n as r varies. ...##
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Star of David and other patterns in the Hosoya-like polynomials triangles
[article]

2017
*
arXiv
*
pre-print

Using this

arXiv:1706.04247v1
fatcat:icapsoqsibgkhhio54utu5p2ui
*generalization*we construct*a*Hosoya-*like**triangle*for polynomials, where its entries are products*of**generalized*Fibonacci polynomials (GFP). ... For every choice*of**a*GFP we obtain*a*triangular array*of*polynomials. In this paper we extend the star*of*David property, also called the Hoggatt-Hansell identity, to this type*of**triangles*. ... In this*triangle*if we replace Fibonacci*numbers**with*the corresponding GFP, we obtain the Hosoya*like*polynomial*triangles*(see Tables 2 and 3 ). ...##
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On a transformation of Riordan moment sequences
[article]

2018
*
arXiv
*
pre-print

It

arXiv:1802.03443v1
fatcat:qkqnt7z4q5hsnmn25qx64yb2se
*associates*lattice path objects*with*permutation objects, and in particular it*associates*the Narayana*triangle**with*the Eulerian*triangle*. ... The ingredients*of*this transformation are series reversion, the Sumudu transform (*a*variant*of*the Laplace transform), and the inverting*of**generating*functions. ... This is the (ordinary) Riordan moment sequence*associated**with*the*family**of*orthogonal polynomials defined by R. ...
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