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Simple closed form Hankel transforms based on the central coefficients of certain Pascal-like triangles
[article]
2006
arXiv
pre-print
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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]
2018
arXiv
pre-print
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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]
2018
arXiv
pre-print
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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
2012
Opuscula Mathematica
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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]
2004
arXiv
pre-print
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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]
2018
arXiv
pre-print
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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
2005
The American mathematical monthly
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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
2005
The American mathematical monthly
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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]
2009
arXiv
pre-print
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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]
2006
arXiv
pre-print
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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]
2020
arXiv
pre-print
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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
2009
The Electronic Journal of Linear Algebra
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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
2016
Journal of the Korean Mathematical Society
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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]
2017
arXiv
pre-print
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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]
2018
arXiv
pre-print
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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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