منابع مشابه
A Pretty Binomial Identity
Elementary proofs abound: the first identity results from choosing x = y = 1 in the binomial expansion of (x+y). The second one may be obtained by comparing the coefficient of x in the identity (1 + x)(1 + x) = (1 + x). The reader is surely aware of many other proofs, including some combinatorial in nature. At the end of the previous century, the evaluation of these sums was trivialized by the ...
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In this article, we demonstrate how the Binomial Theorem in turn arises from a one-parameter generalization of the Sierpinski triangle. The connection between them is given by the sum-of-digits function, s(k), defined as the sum of the digits in the binary representation of k (see [1]). For example, s(3) = s(1·2+1·2) = 2. Towards this end, we begin with a well-known matrix formulation of Sierpi...
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We give an elementary probabilistic proof of a binomial identity. The proof is obtained by computing the probability of a certain event in two different ways, yielding two different expressions for the same quantity. The goal of this note is to give a simple (and interesting) probabilistic proof of the binomial identity n ∑ k=0 ( n k ) (−1) θ θ + k = n ∏ k=1 k θ + k , for all θ > 0 and all n ∈ ...
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In the following we discuss a well-known binomial identity. Many proofs by different methods are known for this identity. Here we present another proof, which uses linear ordinary differential equations of the first order. Several proofs of the well-known identity n ∑ k=0 ( n + k n ) 2 = 2 (1) [4, (1.79)] appear in the literature. In [3, Equation (5.20)], it is proved using partial sums of bino...
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ژورنال
عنوان ژورنال: Social Science Research Network
سال: 2022
ISSN: ['1556-5068']
DOI: https://doi.org/10.2139/ssrn.4097907