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

P. Barry
2006 arXiv   pre-print
We study the Hankel transforms 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.  ... 
arXiv:math/0605169v1 fatcat:szik4tm37ndexoehhe73qowsti

The γ-Vectors of Pascal-like Triangles Defined by Riordan Arrays [article]

Paul Barry
2018 arXiv   pre-print
We are led to the γ-matrices 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.  ... 
arXiv:1804.05027v1 fatcat:ogzkbkdt2zbepklvpxst67bsiy

Counting odd numbers in truncations of Pascal's triangle [article]

Robert G. Donnelly, Molly W. Dunkum, Courtney George, Stefan Schnake
2018 arXiv   pre-print
Presented here is 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.  ... 
arXiv:1807.08181v2 fatcat:xuazxff5ijfd3ko6b7wqz7ehgm

A note on a one-parameter family of non-symmetric number triangles

Maria Irene Falcão, Helmuth R. Malonek
2012 Opuscula Mathematica  
This concerns, in particular, 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.  ... 
doi:10.7494/opmath.2012.32.4.661 fatcat:444jmg7xtbd3tgfb326f4sggvi

The second part of on duality triads' paper [article]

A.K.Kwasniewski
2004 arXiv   pre-print
In this paper Fibonomial 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] .  ... 
arXiv:math/0402288v1 fatcat:74trw42gvnew7aizrtzi6idjpa

Generalized Eulerian Triangles and Some Special Production Matrices [article]

Paul Barry
2018 arXiv   pre-print
Using the previously defined T transform, we 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 .  ... 
arXiv:1803.10297v1 fatcat:2qr7xnz625d4bkst2g327v54yi

Function Series, Catalan Numbers, and Random Walks on Trees

Ibtesam Bajunaid, Joel M. Cohen, Flavia Colonna, David Singman
2005 The American mathematical monthly  
We also thank Glenn Easley for his help 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 .  ... 
doi:10.2307/30037599 fatcat:xjess7f4u5cjbccjnxqre626oe

Function Series, Catalan Numbers, and Random Walks on Trees

Ibtesam Bajunaid, Joel M. Cohen, Flavia Colonna, David Singman
2005 The American mathematical monthly  
We also thank Glenn Easley for his help 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 .  ... 
doi:10.1080/00029890.2005.11920251 fatcat:t7td45feufaexgsi7jcg2ia6cq

Generalized Pascal Triangles and Toeplitz Matrices [article]

A.R. Moghaddamfar, S.M.H. Pooya
2009 arXiv   pre-print
This equality allows us to evaluate 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.  ... 
arXiv:0901.2597v1 fatcat:u7wbe3xajvd2rbwdwuecn6g7ou

Pascal arrays: counting Catalan sets [article]

Robert J Marsh, Paul Martin
2006 arXiv   pre-print
Certain 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.)  ... 
arXiv:math/0612572v1 fatcat:hxs6u6vlqrbn5fdmfehljrffja

On a Central Transform of Integer Sequences [article]

Paul Barry
2020 arXiv   pre-print
Starting from simple sequences 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.  ... 
arXiv:2004.04577v1 fatcat:kyyk34m3urdjti3khjrgdu5uxq

Generalized Pascal triangles and Toeplitz matrices

Ali Reza Moghaddamfar, S. M.H. Pooya
2009 The Electronic Journal of Linear Algebra  
This equality allows for the evaluation 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.  ... 
doi:10.13001/1081-3810.1331 fatcat:wh55um6n7bh5fpgic2udmukqzu

A LINK BETWEEN ORDERED TREES AND GREEN-RED TREES

GI-SANG CHEON, HANA KIM, LOUIS W. SHAPIR
2016 Journal of the Korean Mathematical Society  
The r-ary 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.  ... 
doi:10.4134/jkms.2016.53.1.187 fatcat:ejygx7cnejdtvlv775ryukaoke

Star of David and other patterns in the Hosoya-like polynomials triangles [article]

Rigoberto Florez, Robinson A. Higuita, Antara Mukherjee
2017 arXiv   pre-print
Using this 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 ).  ... 
arXiv:1706.04247v1 fatcat:icapsoqsibgkhhio54utu5p2ui

On a transformation of Riordan moment sequences [article]

Paul Barry
2018 arXiv   pre-print
It 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.  ... 
arXiv:1802.03443v1 fatcat:qkqnt7z4q5hsnmn25qx64yb2se
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