Some Pascal’s like triangles

Document Type : Research Paper

Authors

1 Department of Computer Science, Birjand University of Technology, Birjand, Iran

2 Department of Mathematics, Payame Noor University, P. O. Box 19395-3697, Tehran, Iran.

Abstract

Abstract. In this article, we delve into the fascinating world of numerical triangles similar to Khayyam-Pascal triangle. Our focus is on triangles that are comprised of natural numbers. Along with a combi-natorial interpretation, we also propose an algebraic interpretation for the elements in most cases. We explore in details the triangle of coefficients of Chebyshev polynomials (Chebyshev triangle). Through our analysis, we derive a recursive relation for its elements. This finding sheds new light on the properties of this intriguing numerical construction. To further enhance our understanding of these triangles, we also present new images related to the Catalan, Bell, and Chebyshev triangles. These images provide a clearer visualization of the numerical triangle construction. Overall, this article offers a comprehensive exploration of numerical triangles similar to Khayyam-Pascal triangle and examine some of their properties and relationships for better understanding of these constructions.
 

 
 

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[1] S. M. A. khatami, Beauties of the Khayyam-Pascal triangle, Journal of Mathematics and Society, 5 (2020) 75–92. [In Persian]
[2] M. Mirzavaziri, Count Countables, Ahang Ghalam publisher, 2011. [In Persian]
[3] A. Sabetian, Khayyam-Pascal triangle and some generalizations, Roshd Mathematics Education Journal, 42 (1995) 54–57. [In Persian]
[4] J. Behboodian, M. Bayat and H. Teimoori Faal, Khayyam-Pascal numerical triangle and similar triangles, Sharif University of Technology Publication, 2007. [In Persian]
[5] J. L. Coolidge, The story of the binomial theorem, Amer. Math. Monthly, 56 (1949) 147–157.
[6] R. Rashed, The development of Arabic mathematics: between arithmetic and algebra, (Translated from the 1984 French
original by A. F. W. Armstrong), 156, Kluwer Academic Publishers, Dordrecht, 1994.
[7] N. A. Draim and M. Bicknell, Sums of n-th powers of roots of a given quadratic equation, Fibonacci Quart., 4 (1966)
170–178.
[8] M. Senn, (1,2)-Pascal triangle - OeisWiki, 2016.
[9] H. Belbachir, A. Mehdaoui and L. Szalay, Diagonal sums in Pascal pyramid, J. Combin. Theory Ser. A, 165 (2019) 106–116.
[10] G. E. Andrews, Euler’s “exemplum memorabile inductionis fallacis” and q-trinomial coefficients, J. Amer. Math. Soc., 3 (1990) 653–669.
[11] T. Mansour and M. Schork, Commutation relations, normal ordering, and Stirling numbers, Discrete Mathematics and its Applications (Boca Raton), CRC Press, Boca Raton, FL, 2016.
[12] R. L. Graham, D. E. Knuth and O. Patashnik. Concrete mathematics. A foundation for computer science, Second edition. Addison-Wesley Publishing Company, Reading, MA, 1994.
[13] S. Daboul, J. Mangaldan, M. Z. Spivey and P. J. Taylor, The Lah numbers and the $n$th derivative of $e^{1/x}$, Math. Mag., 86 (2013) 39–47.
[14] T. Kyle Petersen, Eulerian numbers, Birkhäuser Advanced Texts, Birkhäuser/Springer, New York, 2015.
[15] D. F. Bailey, Counting Arrangements of 1’s and -1’s, Math. Mag., 69 (1996) 128–131.
[16] H. S. Wilf, generatingfunctionology, Third edition. A. K. Peters, Ltd., Wellesley, MA, 2006.
[17] W. Eplett, A note about the catalan triangle, Discrete Math., 25 (1979) 289–291.
[18] N. Dershowitz and S. Zaks, Ordered trees and noncrossing partitions, Discrete math., 62 (1986) 215–218.
[19] E. Miller and V. Reiner, Geometric combinatorics, AMS, 2007.
[20] W. Lang, Triangle of coefficients of Chebyshev’s $T(n,x)$ polynomials (powers of x in increasing order), OeisWiki, 2013.
[21] A. Macdougall, A Pascal-like Triangle for Coefficients of Chebyshev Polynomials, The Mathematical Gazette, 83 (1999) 276–280.
[22] G. Dobinski, Summirung der reihe $\sum\frac{n^m}{m!}$ fur $m=1, 2, 3, 4, 5,\ldots$, Grunert’s Archiv, 61 (1877) 333–336.
[23] C. S. Peirce, On the algebra of logic, Amer. J. Math., 3 (1880) 15–57.
[24] A. C. Aitken, A problem in combinations, Edinburgh Mathematical Notes, 28 (1933) xviii-xxiii.
[25] J. H. Conway and R. K. Guy. The book of numbers, Copernicus, New York, 1996.