Relevant bibliographies by topics / Binomial Triangle

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  1. Journal articles
  2. Dissertations / Theses
  3. Books
  4. Book chapters
  5. Conference papers

Academic literature on the topic 'Binomial Triangle'

Author: Grafiati
Published: 2 June 2025
Last updated: 31 July 2025

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Journal articles on the topic "Binomial Triangle"

1

Lewis, Barry. "Generalising Pascal's Triangle." Mathematical Gazette 88, no. 513 (2004): 447–56. http://dx.doi.org/10.1017/s0025557200176089.

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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
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2

Barnes, Benedict, E. D. J. O. Wusu-Ansah, S. K. Amponsah, and I. A. Adjei. "The Proofs of Triangle Inequality Using Binomial Inequalities." European Journal of Pure and Applied Mathematics 11, no. 1 (2018): 352–61. http://dx.doi.org/10.29020/nybg.ejpam.v11i1.3165.

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In this paper, we introduce the different ways of proving the triangle inequality ku − vk ≤ kuk + kvk, in the Hilbert space. Thus, we prove this triangle inequality through the binomial inequality and also, prove it through the Euclidean norm. The first generalized procedure for proving the triangle inequality is feasible for any even positive integer n. The second alternative proof of the triangle inequality establishes the Euclidean norm of any two vectors in the Hilbert space.
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3

M, Mahalakshmi, Kannan J, Deepshika A, and Kaleeswari K. "Existence and Non - Existence of Exponential Diophantine Triangles Over Triangular Numbers." Indian Journal of Science and Technology 16, no. 41 (2023): 3599–604. https://doi.org/10.17485/IJST/v16i41.2338.

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Abstract <strong>Objectives:</strong>&nbsp;The specified problem addressed here is the existence and non-existence of Exponential Diophantine triangles over triangular numbers ( ).&nbsp;<strong>Methods:</strong>&nbsp;An Exponential Diophantine triangle over triangular numbers is defined as a triangle with sides and where and are non - negative integers such that . To prove the existence of such triangles, negative Pell&rsquo;s equation and its solutions are used along with some basic number theoretic concepts. To verify the non-existence, the well-known Catalan&rsquo;s conjecture, binomial exp
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4

M, Mahalakshmi, Kannan J, Deepshika A, and Kaleeswari K. "Existence and Non - Existence of Exponential Diophantine Triangles Over Triangular Numbers." Indian Journal of Science and Technology 16, no. 41 (2023): 3599–604. https://doi.org/10.17485/IJST/v16i41.2338.

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Abstract <strong>Objectives:</strong>&nbsp;The specified problem addressed here is the existence and non-existence of Exponential Diophantine triangles over triangular numbers ( ).&nbsp;<strong>Methods:</strong>&nbsp;An Exponential Diophantine triangle over triangular numbers is defined as a triangle with sides and where and are non - negative integers such that . To prove the existence of such triangles, negative Pell&rsquo;s equation and its solutions are used along with some basic number theoretic concepts. To verify the non-existence, the well-known Catalan&rsquo;s conjecture, binomial exp
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5

M, Mahalakshmi, Kannan J, Deepshika A, and Kaleeswari K. "Existence and Non - Existence of Exponential Diophantine Triangles Over Triangular Numbers." Indian Journal of Science and Technology 16, no. 41 (2023): 3599–604. https://doi.org/10.17485/IJST/v16i41.2338.

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Abstract:
Abstract <strong>Objectives:</strong>&nbsp;The specified problem addressed here is the existence and non-existence of Exponential Diophantine triangles over triangular numbers ( ).&nbsp;<strong>Methods:</strong>&nbsp;An Exponential Diophantine triangle over triangular numbers is defined as a triangle with sides and where and are non - negative integers such that . To prove the existence of such triangles, negative Pell&rsquo;s equation and its solutions are used along with some basic number theoretic concepts. To verify the non-existence, the well-known Catalan&rsquo;s conjecture, binomial exp
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6

M, Mahalakshmi, Kannan J, Deepshika A, and Kaleeswari K. "Existence and Non - Existence of Exponential Diophantine Triangles Over Triangular Numbers." Indian Journal of Science and Technology 16, no. 41 (2023): 3599–604. https://doi.org/10.17485/IJST/v16i41.2338.

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Abstract:
Abstract <strong>Objectives:</strong>&nbsp;The specified problem addressed here is the existence and non-existence of Exponential Diophantine triangles over triangular numbers ( ).&nbsp;<strong>Methods:</strong>&nbsp;An Exponential Diophantine triangle over triangular numbers is defined as a triangle with sides and where and are non - negative integers such that . To prove the existence of such triangles, negative Pell&rsquo;s equation and its solutions are used along with some basic number theoretic concepts. To verify the non-existence, the well-known Catalan&rsquo;s conjecture, binomial exp
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7

M, Mahalakshmi, Kannan J, Deepshika A, and Kaleeswari K. "Existence and Non - Existence of Exponential Diophantine Triangles Over Triangular Numbers." Indian Journal of Science and Technology 16, no. 41 (2023): 3599–604. https://doi.org/10.17485/IJST/v16i41.2338.

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Abstract:
Abstract <strong>Objectives:</strong>&nbsp;The specified problem addressed here is the existence and non-existence of Exponential Diophantine triangles over triangular numbers ( ).&nbsp;<strong>Methods:</strong>&nbsp;An Exponential Diophantine triangle over triangular numbers is defined as a triangle with sides and where and are non - negative integers such that . To prove the existence of such triangles, negative Pell&rsquo;s equation and its solutions are used along with some basic number theoretic concepts. To verify the non-existence, the well-known Catalan&rsquo;s conjecture, binomial exp
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8

Ahmia, Moussa, and Hacène Belbachir. "Preserving log-concavity for p,q-binomial coefficient." Discrete Mathematics, Algorithms and Applications 11, no. 02 (2019): 1950017. http://dx.doi.org/10.1142/s1793830919500174.

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We study the log-concavity of a sequence of [Formula: see text]-binomial coefficients located on a ray of the [Formula: see text]-Pascal triangle for certain directions, and we establish the preserving log-concavity of linear transformations associated to [Formula: see text]-Pascal triangle.
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9

Zhu, Bao-Xuan. "Log-concavity and strong q-log-convexity for Riordan arrays and recursive matrices." Proceedings of the Royal Society of Edinburgh: Section A Mathematics 147, no. 6 (2017): 1297–310. http://dx.doi.org/10.1017/s0308210516000500.

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Let [An,k]n,k⩾0 be an infinite lower triangular array satisfying the recurrencefor n ⩾ 1 and k ⩾ 0, where A0,0 = 1, A0,k = Ak,–1 = 0 for k &gt; 0. We present some criteria for the log-concavity of rows and strong q-log-convexity of generating functions of rows. Our results can be applied to many well-known triangular arrays, such as the Pascal triangle, the Stirling triangle of the second kind, the Bell triangle, the large Schröder triangle, the Motzkin triangle, and the Catalan triangles of Aigner and Shapiro, in a unified approach. In addition, we prove that the binomial transformation not o
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10

Beiu, Valeriu, Leonard Dăuş, Marilena Jianu, Adela Mihai, and Ion Mihai. "On a Surface Associated with Pascal’s Triangle." Symmetry 14, no. 2 (2022): 411. http://dx.doi.org/10.3390/sym14020411.

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An open problem in reliability theory is that of finding all the coefficients of the reliability polynomial associated with particular networks. Because reliability polynomials can be expressed in Bernstein form (hence linked to binomial coefficients), it is clear that an extension of the classical discrete Pascal’s triangle (comprising all the binomial coefficients) to a continuous version (exhibiting infinitely many values in between the binomial coefficients) might be geometrically helpful and revealing. That is why we have decided to investigate the geometric properties of a continuous ext
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Dissertations / Theses on the topic "Binomial Triangle"

1

Riderer, Lucia. "Numbers of generators of ideals in local rings and a generalized Pascal's Triangle." CSUSB ScholarWorks, 2005. https://scholarworks.lib.csusb.edu/etd-project/2732.

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This paper defines generalized binomial coefficients and shows that they can be used to generate generalized Pascal's Triangles and have properties analogous to binomial coefficients. It uses the generalized binomial coefficients to compute the Dilworth number and the Sperner number of certain rings.
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2

Silva, Salatiel Dias da. "Estudo do binômio de Newton." Universidade Federal da Paraíba, 2013. http://tede.biblioteca.ufpb.br:8080/handle/tede/7526.

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Submitted by Viviane Lima da Cunha (viviane@biblioteca.ufpb.br) on 2015-10-16T15:00:20Z No. of bitstreams: 1 arquivototal.pdf: 971519 bytes, checksum: 75c5acddc58c0f9e43eb4d646a3fa8fd (MD5)<br>Approved for entry into archive by Maria Suzana Diniz (msuzanad@hotmail.com) on 2015-10-16T22:38:36Z (GMT) No. of bitstreams: 1 arquivototal.pdf: 971519 bytes, checksum: 75c5acddc58c0f9e43eb4d646a3fa8fd (MD5)<br>Made available in DSpace on 2015-10-16T22:38:36Z (GMT). No. of bitstreams: 1 arquivototal.pdf: 971519 bytes, checksum: 75c5acddc58c0f9e43eb4d646a3fa8fd (MD5) Previous issue date: 2013-08-1
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3

Silva, MÃrcio RebouÃas da. "NÃmeros binomiais: uma abordagem combinatÃria para o ensino mÃdio." Universidade Federal do CearÃ, 2015. http://www.teses.ufc.br/tde_busca/arquivo.php?codArquivo=15115.

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Este trabalho tem por finalidade apresentar uma abordagem, para o Ensino MÃdio, de nÃmeros binomiais (incluindo as propriedades do triÃngulo de Pascal e binÃmio de Newton), contendo as demonstraÃÃes combinatÃrias, ao utilizar dupla contagem, juntamente com as demonstraÃÃes algÃbricas, como parcialmente jà à feito, alÃm de generalizar, citando os nÃmeros trinomiais (incluindo as propriedades da pirÃmide de Pascal) e os nÃmeros multinomiais (incluindo o polinÃmio de Leibniz).<br>This project aims at presenting an approach of binomial numbers for high school (including Pascalâs triangle propertie
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4

Montrezor, Camila Lopes. "Funções aritméticas." Universidade de São Paulo, 2017. http://www.teses.usp.br/teses/disponiveis/55/55136/tde-25072017-082655/.

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Neste estudo, apresentamos conteúdos matemáticos adaptáveis tanto para os anos finais do ensino fundamental quanto para o ensino médio. Iniciamos com um conjunto de ideias preliminares: indução matemática, triângulo de Pascal, Binômio de Newton e relações trigonométricas, para a obtenção de fórmulas de somas finitas, em que os valores das parcelas são computados sobre números inteiros consecutivos, e da técnica de transformação de soma finita em telescópica. Enunciamos Progressões Aritméticas e Geométricas como sequências numéricas e suas propriedades, obtendo a soma de seus n primeiros termos
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Books on the topic "Binomial Triangle"

1

Cai, Zongxi. Deng zhou wen ti. Ke xue chu ban she, 2002.

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2

Edwards, A. W. F. Pascal's Arithmetical Triangle: The Story of a Mathematical Idea. Dover Publications, Incorporated, 2019.

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3

Edwards, A. W. F. Pascal's Arithmetical Triangle: The Story of a Mathematical Idea. Dover Publications, Incorporated, 2019.

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Book chapters on the topic "Binomial Triangle"

1

Lovász, L., J. Pelikán, and K. Vesztergombi. "Binomial Coefficients and Pascal’s Triangle." In Discrete Mathematics. Springer New York, 2003. http://dx.doi.org/10.1007/0-387-21777-0_3.

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2

Vishwakarma, Suryaprakash, and Seema Purohit. "RASCAL TRIANGLE." In Futuristic Trends in Contemporary Mathematics & Applications Volume 3 Book 2. Iterative International Publishers, Selfypage Developers Pvt Ltd, 2024. http://dx.doi.org/10.58532/v3bkcm2p6ch2.

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In Combinatorics use of Pascal Triangle techniques and identities is well known when it comes to deriving Binomial Coefficients, Fibonacci Numbers, Interesting Numbers Patterns. Literature shows though not much efforts are being done to generate the Pascal like pattens, a decade ago in 2010, Pascal like triangles are generated by three middle school students using their own alternate ways, without knowing what Pascal Triangle is about. They coined the name “Rascal Triangle” for the new number patterns generated by them. In this Chapter, researchers tried to regenerate the generalizations of Ra
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3

"Binomial Coefficients and Pascal’s Triangle." In Discrete Mathematics with Ducks. Chapman and Hall/CRC, 2018. http://dx.doi.org/10.1201/9781315167671-22.

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4

"Binomial Coefficients and Pascal’s Triangle." In Student Handbook for Discrete Mathematics with Ducks. A K Peters/CRC Press, 2015. http://dx.doi.org/10.1201/b18720-11.

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5

"Pascal’s triangle and the binomial theorem." In Fat Chance. Cambridge University Press, 2019. http://dx.doi.org/10.1017/9781108610278.009.

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6

Wilson, Robin. "3. Permutations and combinations." In Combinatorics: A Very Short Introduction. Oxford University Press, 2016. http://dx.doi.org/10.1093/actrade/9780198723493.003.0003.

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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, factor
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Osterlind, Steven J. "The Patterns of Large Numbers." In The Error of Truth. Oxford University Press, 2019. http://dx.doi.org/10.1093/oso/9780198831600.003.0004.

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This chapter advances the historical context for quantification by describing the climate of the day—social, cultural, political, and intellectual—as fraught with disquieting influences. Forces leading to the French Revolution were building, and the colonists in America were fighting for secession from England. During this time, three important number theorems came into existence: the binomial theorem, the law of large numbers, and the central limit theorem. Each is described in easy-to-understand language. These are fundamental to how numbers operate in a probability circumstance. Pascal’s tr
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Conference papers on the topic "Binomial Triangle"

1

Elfeky, Assem H., Ahmed F. Miligy, Hassan Nadir Kheirallah, and Mohamed R. M. Rizk. "Analysis and Implementation of Binomial SIW Slot Antenna Array Based on Pascal’s Triangle Using 6-Port SIW Directional Coupler for X-Band Systems." In 2022 International Telecommunications Conference (ITC-Egypt). IEEE, 2022. http://dx.doi.org/10.1109/itc-egypt55520.2022.9855694.

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