Generalising Pascal's Triangle

2004 ◽  
Vol 88 (513) ◽  
pp. 447-456 ◽  
Author(s):  
Barry Lewis
Keyword(s):  
Prime Number ◽  
Recurrence Relation ◽  
Stirling Numbers ◽  
Binomial Coefficients ◽  
Combinatorial Properties ◽  
Pascal’S Triangle ◽  
Binomial Expansion ◽  
Pascal Triangle ◽  
Divisibility Property ◽  
Pascal's Triangle

Pascal's triangle, the Binomial expansion and the recurrence relation between its entries are all inextricably linked. In the normal course of events, the Binomial expansion leads to Pascal's Triangle, and thence to the recurrence relation between its entries. In this article we are going to reverse this process to make it possible to explore a particular type of generalisation of such interlinked structures, by generalising the recurrence relation and then exploring the resulting generalised ‘Pascal Triangle’ and ‘Binomial expansion’. Within the spectrum of generalisations considered, we find exactly four of particular significance: those concerned with the Binomial coefficients, the Stirling numbers of both kinds, and a lesser known set of numbers – the Lah numbers. We also examine the combinatorial properties of the entries in these triangles and a prime number divisibility property that they all share. Thereby, we achieve a remarkable synthesis of these different entities.

scholarly journals On Unimodality Problems in Pascal's Triangle

10.37236/837 ◽  
2008 ◽  
Vol 15 (1) ◽  
Author(s):  
Xun-Tuan Su ◽  
Yi Wang
Keyword(s):  
Affirmative Answer ◽  
Binomial Coefficients ◽  
Pascal’S Triangle ◽  
Pascal Triangle ◽  
Pascal's Triangle

Many sequences of binomial coefficients share various unimodality properties. In this paper we consider the unimodality problem of a sequence of binomial coefficients located in a ray or a transversal of the Pascal triangle. Our results give in particular an affirmative answer to a conjecture of Belbachir et al which asserts that such a sequence of binomial coefficients must be unimodal. We also propose two more general conjectures.

scholarly journals Partial factorizations of products of binomial coefficients

2021 ◽  
pp. 1-38
Author(s):  
Lara Du ◽  
Jeffrey C. Lagarias
Keyword(s):  
Prime Number ◽  
Riemann Hypothesis ◽  
Remainder Term ◽  
Prime Number Theorem ◽  
Product Formula ◽  
Binomial Coefficients ◽  
Limit Function ◽  
Pascal’S Triangle ◽  
Prime Factors ◽  
Pascal's Triangle

Let [Formula: see text] the product of the elements of the [Formula: see text]th row of Pascal’s triangle. This paper studies the partial factorizations of [Formula: see text] given by the product [Formula: see text] of all prime factors [Formula: see text] of [Formula: see text] having [Formula: see text], counted with multiplicity. It shows [Formula: see text] as [Formula: see text] for a limit function [Formula: see text] defined for [Formula: see text]. The main results are deduced from study of functions [Formula: see text] that encode statistics of the base [Formula: see text] radix expansions of the integer [Formula: see text] (and smaller integers), where the base [Formula: see text] ranges over primes [Formula: see text]. Asymptotics of [Formula: see text] and [Formula: see text] are derived using the prime number theorem with remainder term or conditionally on the Riemann hypothesis.

scholarly journals Pascal type properties of Betti numbers

1994 ◽  
Vol 17 (3) ◽  
pp. 545-552
Author(s):  
Tilak de Alwis
Keyword(s):  
Polynomial Ring ◽  
Betti Numbers ◽  
Binomial Coefficients ◽  
Minimal Resolution ◽  
Pascal’S Triangle ◽  
Specific Formula ◽  
Pascal's Triangle

In this paper, we will describe the Pascal Type properties of Betti numbers of ideals associated ton-gons. These are quite similar to the properties enjoyed by the Pascal's Triangle, concerning the binomial coefficients. By definition, the Betti numbersβt(n)of an idealIassociated to ann-gon are the ranks of the modules in a free minimal resolution of theR-moduleR/I, whereRis the polynomial ringk[x1,x2,…,xn]. Herekis any field andx1,x2,…,xnare indeterminates. We will prove those properties using a specific formula for the Betti numbers.

scholarly journals PASCAL TRIANGLE AND PYTHAGOREAN TRIPLES

2021 ◽  
Vol 8 (8) ◽  
pp. 75-80
Author(s):  
R. Sivaraman
Keyword(s):  
Pythagorean Triples ◽  
Pascal’S Triangle ◽  
Pascal Triangle ◽  
Pascal's Triangle

The concept of Pascal’s triangle has fascinated mathematicians for several centuries. Similarly, the idea of Pythagorean triples prevailing for more than two millennia continue to surprise even today with its abundant properties and generalizations. In this paper, I have demonstrated ways through four theorems to determine Pythagorean triples using entries from Pascal’s triangle.

scholarly journals Recurrence Relations for the Linear Transformation Preserving the Strong $q$-Log-Convexity

10.37236/5913 ◽  
2016 ◽  
Vol 23 (3) ◽  
Author(s):  
Lily Li Liu ◽  
Ya-Nan Li
Keyword(s):  
Recurrence Relation ◽  
Linear Transformation ◽  
Recurrence Relations ◽  
Stirling Numbers ◽  
Binomial Coefficients ◽  
Sufficient Condition ◽  
Linear Transformations ◽  
Whitney Numbers ◽  
Three Term Recurrence Relation

Let $[T(n,k)]_{n,k\geqslant0}$ be a triangle of positive numbers satisfying the three-term recurrence relation\[T(n,k)=(a_1n+a_2k+a_3)T(n-1,k)+(b_1n+b_2k+b_3)T(n-1,k-1).\]In this paper, we give a new sufficient condition for linear transformations\[Z_n(q)=\sum_{k=0}^{n}T(n,k)X_k(q)\]that preserves the strong $q$-log-convexity of polynomials sequences. As applications, we show linear transformations, given by matrices of the binomial coefficients, the Stirling numbers of the first kind and second kind, the Whitney numbers of the first kind and second kind, preserving the strong $q$-log-convexity in a unified manner.

3. Permutations and combinations

2016 ◽  
pp. 28-49
Author(s):  
Robin Wilson
Keyword(s):  
Binomial Coefficients ◽  
Combination Rules ◽  
Binomial Theorem ◽  
Breakfast Cereals ◽  
Pascal’S Triangle ◽  
Pascal's Triangle

Permutations and combinations have been studied for thousands of years. ‘Permutations and combinations’ considers selecting objects from a collection, either in a particular order (such as when ranking breakfast cereals) or without concern for order (such as when dealing out a bridge hand). It describes and investigates four types of selection—ordered selections with repetition, ordered selections without repetition, unordered selections without repetition, and unordered selections with repetition—and shows how they are related to permutations, combinations, the three combination rules, factorials, Pascal’s triangle, the binomial theorem, binomial coefficients, and distributions.

More power to Pascal

2008 ◽  
Vol 92 (525) ◽  
pp. 454-465 ◽  
Author(s):  
Barry Lewis
Keyword(s):  
Fibonacci Numbers ◽  
Ancient Greece ◽  
Binomial Coefficients ◽  
Medieval Italy ◽  
Pascal’S Triangle ◽  
Pascal's Triangle

Pascal’s triangle is the most famous of all number arrays - full of patterns and surprises. One surprise is the fact that lurking amongst these binomial coefficients are the triangular and pyramidal numbers of ancient Greece, the combinatorial numbers which arose in the Hindu studies of arrangements and selections, together with the Fibonacci numbers from medieval Italy. New identities continue to be discovered, so much so that their publication frequently excites no one but the discoverer.

Sign in / Sign up

Export Citation Format

Share Document