Binomial Coefficient Formulas by General Reasoning

1968 ◽  
Vol 61 (4) ◽  
pp. 399-402
Author(s):  
Jack M. Elkin
Keyword(s):  
Binomial Coefficient ◽  
Algebraic Manipulation ◽  
Binomial Coefficients ◽  
Binomial Theorem ◽  
Relative Ease ◽  
Methods Of Analysis ◽  
Summation Of Series ◽  
Mathematical Formulas ◽  
High Degree ◽  
Pascal's Triangle

The binomial coefficients are an almost endless source of formulas for the summation of series. A reference to the “Problems for Solution” pages of the American Mathematical Monthly or to an advanced collection of mathematical formulas will convince anyone who has not yet discovered this for himself. Some of these series summations can be derived with relative ease with the help of the binomial theorem or Pascal's triangle; many require a high degree of virtuosity in algebraic manipulation and, often, advanced methods of analysis. A number of them can be obtained simply by reasoning logically about the meaning of certain combinatorial expressions, with recourse to only a minimum of algebra or to none at all. These, naturally, have a special appeal of their own, and it is the purpose of this article to illustrate several such derivations.

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.

scholarly journals Analogues of the binomial coefficient theorems of Gauss and Jacobi

2016 ◽  
Vol 12 (08) ◽  
pp. 2125-2145
Author(s):  
Abdullah Al-Shaghay ◽  
Karl Dilcher
Keyword(s):  
Gamma Function ◽  
Simple Form ◽  
Binomial Coefficient ◽  
Catalan Numbers ◽  
Binomial Coefficients

The theorems of Gauss and Jacobi that give modulo [Formula: see text] evaluations of certain central binomial coefficients have been extended, since the 1980s, to more classes of binomial coefficients and to congruences modulo [Formula: see text]. In this paper, we further extend these results to congruences modulo [Formula: see text]. In the process, we prove congruences to arbitrarily high powers of [Formula: see text] for certain quotients of Gauss factorials that resemble binomial coefficients and are related to Morita's [Formula: see text]-adic gamma function. These congruences are of a simple form and involve Catalan numbers as coefficients.

Preserving log-concavity for p,q-binomial coefficient

2019 ◽  
Vol 11 (02) ◽  
pp. 1950017
Author(s):  
Moussa Ahmia ◽  
Hacène Belbachir
Keyword(s):  
Binomial Coefficient ◽  
Binomial Coefficients ◽  
Linear Transformations ◽  
Pascal Triangle

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.

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 A Generalized Inverse Binomial Summation Theorem and Some Hypergeometric Transformation Formulas

2016 ◽  
Vol 2016 ◽  
pp. 1-14
Author(s):  
S. M. Ripon
Keyword(s):  
Generalized Inverse ◽  
Binomial Coefficient ◽  
Bell Polynomials ◽  
Binomial Theorem ◽  
Summation Theorem ◽  
Transformation Formulas

A generalized binomial theorem is developed in terms of Bell polynomials and by applying this identity some sums involving inverse binomial coefficient are calculated. A technique is derived for calculating a class of hypergeometric transformation formulas and also some curious q series identities.

Analysis of Self-Excited Induction Generators Using Symbolic Programming

1992 ◽  
Vol 29 (4) ◽  
pp. 329-338 ◽  
Author(s):  
T. F. Chan
Keyword(s):  
Numerical Analysis ◽  
Programming Language ◽  
Computer Programming ◽  
Algebraic Manipulation ◽  
The Self ◽  
Induction Generator ◽  
Straightforward Manner ◽  
Symbolic Programming ◽  
Induction Generators ◽  
High Degree

Analysis of self-excited induction generators using symbolic programming Using the symbolic programming language MACSYMA, the self-excited induction generator may be analysed in a straightforward manner with a high degree of accuracy. Very little manual effort need be spent on algebraic manipulation, numerical analysis and computer programming. Typical program sessions are cited to illustrate the elegance of this approach.

scholarly journals On certain definite integrals. No 14

1886 ◽  
Vol 39 (239-241) ◽  
pp. 22-23
Keyword(s):  
Binomial Coefficients ◽  
Definite Integrals ◽  
Summation Of Series

It follows from the expansion of cos n θ in terms of the cosines of the multiples of θ , that n . n - 1/2. n - 2/3 ... n - r +1/ r = 2 n / π ∫ π 0 cos nθ cos ( n -2 r θdθ , and consequently this theorem can be used in the summation of series involving binomial coefficients. I propose to give a few examples of this.

Speaking style as an expression of solidarity: Words per pause

1990 ◽  
Vol 19 (1) ◽  
pp. 81-88 ◽  
Author(s):  
Norman Markel
Keyword(s):  
Nonverbal Communication ◽  
Expressive Language ◽  
Practical Implication ◽  
Diagnostic Device ◽  
Relative Ease ◽  
Significant Difference ◽  
Litmus Test ◽  
High Degree ◽  
Speaking Style ◽  
Statistical Results

ABSTRACTThis study examines the use of words per pause (W/P) as a practical means for identifying solidarity in everyday conversation. Eight listeners recorded the narratives of a female and a male, either friends or strangers. Ten speakers were categorized as friends and six as strangers; they talked about a good and a bad experience. Average reliability of coding pauses was .83. The results indicated a statistically significant difference in W/P of speakers who were friends and those who were strangers. Statistical results support the conclusion that friends are more likely to employ many W/P and strangers few W/P. One practical implication of this study is that W/P can be employed by researchers with relative ease and a high degree of reliability for investigations of speaking style in a variety of contexts. A second practical implication is that W/P is a diagnostic device that can serve as a social litmus test in everyday conversation to identify the expression of sympathy and estrangement. (Expressive language, nonverbal communication, paralanguage, pauses, psycholinguistics, sociolinguistics, solidarity, speech and personality)

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