Applicable Analysis and Discrete Mathematics 2022 Volume 16, Issue 2, Pages: 328-349
https://doi.org/10.2298/AADM200411028A
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Asymmetric extension of Pascal-Delannoy triangles
Amrouche Said (USTHB, Faculty of Mathematics, RECITS Laboratory, Bab Ezzouar, Algiers, Algeria), saidamrouchee@gmail.com
Belbachir Hacène (Scientific and Technical Information Research Center, Ben Aknoun, Algiers, Algeria), hbelbachir@usthb.dz
In this paper, we give a generalization of the Pascal triangle called the
quasi s-Pascal triangle. For this, consider a set of lattice path, which is
a dual approach to the definition of Ramirez and Sirvent: A Generalization
of the k-bonacci Sequence from Riordan Arrays. The electronic journal of
combinatorics, 22(1) (2015), 1-38. We give the recurrence relation for the
sum of elements lying over finite ray of the quasi s-Pascal triangle, then,
we establish a q-analogue of the coefficient of this triangle. Some
identities are also given.
Keywords: Generalized Pascal triangle, Recurrence relations, Generating function, q-analogue
Show references
K. Alladi, V. E. Hoggatt Jr: On tribonacci numbers and related functions. Fibonacci Quart, 15(1) (1977), 42-45.
S. Amrouche, H. Belbachir: Unimodality and linear recurrences associated with rays in the Delannoy triangle. Turkish Journal of Mathematics, 44(1) (2020), 118-130.
S. Amrouche, H. Belbachir, and J. L. Ramirez: Unimodality, linear recurrences and combinatorial properties associated to rays in the generalized Delannoy matrix. Journal of Difference Equations and Applications, 25(8) (2019), 1200-1215.
G. Andrews, J. Baxter: Lattice gas generalization of the hard hexagon model III q-trinomials coefficients. J Stat Phys, 47 (1987), 297-330.
P. Barry: On integer-sequence-based constructions of generalized Pascal triangles. Journal of Integer Sequences, 9 (2006), Article 06.2.4.
H. Belbachir, A. Benmezai: A q-analogue for bisnomial coefficients and generalized Fibonacci sequences. C. R. Math. Acad. Sci. Paris, 352(3) (2014), 167-171.
H. Belbachir, S. Bouroubi, A. Khelladi: Connection between ordinary multinomials, Fibonacci numbers, Bell polynomials and discrete uniform distribution. Annales Mathematicae et Informaticae. 35(2008), 21-30.
H. Belbachir, T.Komatsu, L. Szalay: Linear recurrences associated to rays in Pascal’s triangle and combinatorial identities. Mathematica Slovaca 64 (2) (2014), 287-300.
H.Belbachir, L. Szalay: On the arithmetic triangles. Siauliai Math. Sem, 9(17) (2014), 15-26.
A. Benmezai:Ph.D Thesis http://theses.univ-oran1.dz/these.php?id=16201630t.
R. Bollinger: A note on Pascal T-triangles, Multinomial coefficients and Pascal pyramids: Fibonacci Quart, 24 (1986), 140-144.
S. Butler, P. Karasik: A note on nested sums. J. Integer Seq, 13(2) (2010), 3.
L. Carlitz: Fibonacci Notes-3 q-Fibonacci Numbers. Fibonacci Quart, 12 (1974) 317-322.
J. Cigler: A new class of q-Fibonacci polynomials. The electronic journal of combinatorics, 10(1) (2003), 19.
J. Cigler: q-Fibonacci polynomials. Fibonacci Quart, 41(1) (2003), 31-40.
M. Dziemianczuk: Counting lattice paths with four types of steps. Graphs Combin, 30 (2014), 1427-1452.
R. D. Fray, D. P. Roselle: Weighted lattice paths. Pacific J. Math, 37(1) (1971), 85-96.
S. G. Mohanty, B. R. Handa: On lattice paths with several diagonal steps. Canadian Math. Bull, 11 (1968), 537-545.
A. de Moivre: The doctrine of chances, Third edition 1756 (first ed. 1718 and second ed. (1738)), reprinted by Chelsea, N. Y. (1967).
A. de Moivre: Miscellanca Analytica de Scrichus et Quadraturis. J. Tomson and J. Watts, London, (1731).
J. L. Ramirez, V. F. Sirvent: A Generalization of the k-bonacci Sequence from Riordan Arrays. The electronic journal of combinatorics, 22(1) (2015), 1-38.
V. K. Rohatgi: A note on lattice paths with diagonal steps. Canadian Math. Bull, 7 (1964), 470-472.