Journal of Integer Sequences, Vol. 25 (2022), Article 22.7.1

On Two Families of Generalizations of Pascal's Triangle

Michael A. Allen and Kenneth Edwards
Physics Department
Faculty of Science
Mahidol University
Rama 6 Road
Bangkok 10400


We consider two families of Pascal-like triangles that have all ones on the left side and ones separated by m – 1 zeros on the right side. The m = 1 cases are Pascal's triangle and the two families also coincide when m = 2. Members of the first family obey Pascal's recurrence everywhere inside the triangle. We show that the m-th triangle can also be obtained by reversing the elements up to and including the main diagonal in each row of the (1/(1 – xm), x/(1 – x)) Riordan array. Properties of this family of triangles can be obtained quickly as a result. The (n, k)-th entry in the m-th member of the second family of triangles is the number of tilings of an (n + k) × 1 board that use k (1, m – 1)-fences and nk unit squares. A (1, g)-fence is composed of two unit square sub-tiles separated by a gap of width g. We show that the entries in the antidiagonals of these triangles are coefficients of products of powers of two consecutive Fibonacci polynomials and give a bijective proof that these coefficients give the number of k-subsets of {1, 2, ... , nm} such that no two elements of a subset differ by m. Other properties of the second family of triangles are also obtained via a combinatorial approach. Finally, we give necessary and sufficient conditions for any Pascal-like triangle (or its row-reversed version) derived from tiling (n × 1)-boards to be a Riordan array.

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(Concerned with sequences A000045 A000079 A000217 A001045 A006324 A006498 A007318 A011973 A059259 A077947 A079962 A115451 A118923 A123521 A157897 A335964 A349839 A349840 A349841 A349842 A349843 A350110 A350111 A350112.)

Received January 18 2022; revised versions received May 26 2022; June 21 2022. Published in Journal of Integer Sequences, July 29 2022.

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