Shorter Notes: Simple Proof That a p-ADIC Pascal's Triangle is 120 ∘ Rotatable

1976 ◽  
Vol 59 (2) ◽  
pp. 406
Author(s):  
Sin Hitotumatu ◽  
Daihachiro Sato
Keyword(s):  
Simple Proof ◽  
Pascal’S Triangle ◽  
Pascal's Triangle

scholarly journals Infinite log-concavity: developments and conjectures

2009 ◽  
Vol DMTCS Proceedings vol. AK,... (Proceedings) ◽  
Author(s):  
Peter R. W. McNamara ◽  
Bruce E. Sagan
Keyword(s):  
Recent Result ◽  
Simple Proof ◽  
Symmetric Functions ◽  
Toeplitz Matrices ◽  
Real Numbers ◽  
Pascal’S Triangle ◽  
International Audience ◽  
Pascal's Triangle ◽  
Nonnegative Sequence ◽  
Sequence Call

International audience Given a sequence $(a_k)=a_0,a_1,a_2,\ldots$ of real numbers, define a new sequence $\mathcal{L}(a_k)=(b_k)$ where $b_k=a_k^2-a_{k-1}a_{k+1}$. So $(a_k)$ is log-concave if and only if $(b_k)$ is a nonnegative sequence. Call $(a_k)$ $\textit{infinitely log-concave}$ if $\mathcal{L}^i(a_k)$ is nonnegative for all $i \geq 1$. Boros and Moll conjectured that the rows of Pascal's triangle are infinitely log-concave. Using a computer and a stronger version of log-concavity, we prove their conjecture for the $n$th row for all $n \leq 1450$. We can also use our methods to give a simple proof of a recent result of Uminsky and Yeats about regions of infinite log-concavity. We investigate related questions about the columns of Pascal's triangle, $q$-analogues, symmetric functions, real-rooted polynomials, and Toeplitz matrices. In addition, we offer several conjectures. Étant donné une suite $(a_k)=a_0,a_1,a_2,\ldots$ de nombres réels, on définit une nouvelle suite $\mathcal{L}(a_k)=(b_k)$ où $b_k=a_k^2-a_{k-1}a_{k+1}$. Alors $(a_k)$ est log-concave si et seulement si $(b_k)$ est une suite non négative. On dit que $(a_k)$ est $\textit{infiniment log-concave}$ si $\mathcal{L}^i(a_k)$ est non négative pour tout $i \geq 1$. Boros et Moll ont conjecturé que les lignes du triangle de Pascal sont infiniment log-concave. Utilisant un ordinateur et une version plus forte de log-concavité, on vérifie leur conjecture pour la $n$ième ligne, pour tout $n \leq 1450$. On peut aussi utiliser nos méthodes pour donner une preuve simple d'un résultat récent de Uminsky et Yeats à propos des régions de log-concavité infini. Reliées à ces idées, on examine des questions à propos des colonnes du triangle de Pascal, des $q$-analogues, des fonctions symétriques, des polynômes avec racines réelles, et des matrices de Toeplitz. De plus, on offre plusieurs conjectures.

Pascal's Prism

2012 ◽  
Vol 96 (536) ◽  
pp. 213-220
Author(s):  
Harlan J. Brothers
Keyword(s):  
Probability Theory ◽  
Fractal Geometry ◽  
Euclidean Geometry ◽  
Fibonacci Series ◽  
Pascal’S Triangle ◽  
Number Sequences ◽  
Definition Of ◽  
Image Position ◽  
Pascal's Triangle

Pascal's triangle is well known for its numerous connections to probability theory [1], combinatorics, Euclidean geometry, fractal geometry, and many number sequences including the Fibonacci series [2,3,4]. It also has a deep connection to the base of natural logarithms, e [5]. This link to e can be used as a springboard for generating a family of related triangles that together create a rich combinatoric object.2. From Pascal to LeibnizIn Brothers [5], the author shows that the growth of Pascal's triangle is related to the limit definition of e.Specifically, we define the sequence sn; as follows [6]:

3057. A Note on Pascal's Triangle

1963 ◽  
Vol 47 (359) ◽  
pp. 57
Author(s):  
Robert Croasdale
Keyword(s):  
Pascal’S Triangle ◽  
Pascal's Triangle

scholarly journals The Fibonacci p-numbers and Pascal’s triangle

2016 ◽  
Vol 3 (1) ◽  
pp. 1264176 ◽  
Author(s):  
Kantaphon Kuhapatanakul ◽  
Lishan Liu
Keyword(s):  
Pascal’S Triangle ◽  
Pascal's Triangle

scholarly journals Arithmetic Triangle

2017 ◽  
Vol 9 (2) ◽  
pp. 100
Author(s):  
Luis Dias Ferreira
Keyword(s):  
Arithmetic Progression ◽  
Modus Operandi ◽  
Interesting Point ◽  
Pascal’S Triangle ◽  
Initial Term ◽  
The Common ◽  
Pascal's Triangle

The product of the first $n$ terms of an arithmetic progression may be developed in a polynomial of $n$ terms. Each one of them presents a coefficient $C_{nk}$ that is independent from the initial term and the common difference of the progression. The most interesting point is that one may construct an "Arithmetic Triangle'', displaying these coefficients, in a similar way one does with Pascal's Triangle. Moreover, some remarkable properties, mainly concerning factorials, characterize the Triangle. Other related `triangles' -- eventually treated as matrices -- also display curious facts, in their linear \emph{modus operandi}, such as successive "descendances''.

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