Journal of Integer Sequences, Vol. 28 (2025), Article 25.3.5

The 2-Pascal Triangle and a Related Riordan Array


Yassine Otmani
Faculty of Mathematics
USTHB, RECITS Laboratory
P.O. Box 32
El Alia, 16111
Bab Ezzouar
Algiers
Algeria

Abstract:

In this paper, we determine the structure of the 2-Pascal triangle and prove that it consists of even rows within a specific proper Riordan array. We provide the generating functions for two identities involving the coefficients of the triangle. As a consequence, we derive some combinatorial identities, including the generating functions for the sums of diagonal elements along a finite ray through the 2-Pascal triangle. Finally, we determine the inverse transform of the trinomial transform.


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(Concerned with sequences A000032 A000073 A000129 A001333 A001644 A002605 A008277 A008287 A027907 A027914 A035343 A052948 A063260 A075115 A077828 A077835 A077843 A077846 A077943 A081179 A081336 A083878 A083882 A094531 A097893 A099463 A102001 A103770 A103771 A133872 A146559 A152107 and A246437.)


Received September 17 2023; revised versions received November 9 2024; November 16 2024; December 3 2024. Published in Journal of Integer Sequences, April 19 2025.


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