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 note we shall prove the following curious identity of sums of powers of the partial sum of binomial coefficients.
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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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for any a, b, c ∈ N = {0, 1, 2, . . .}. During the second author’s visit (January–March, 2005) to the Institute of Camille Jordan at Univ. Lyon-I, Dr. Victor Jun Wei Guo told Sun his following conjecture involving sums of products of three binomial coefficients and said that he failed to prove this “difficult conjecture” during the past two years though he had tried to work it out again and again.
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ژورنال
عنوان ژورنال: Discrete Mathematics
سال: 1996
ISSN: 0012-365X
DOI: 10.1016/0012-365x(95)00086-c