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Category Archives: binomial coefficients
Pascal Matrices and Binomial Inversion
In this post we’ll look at the relationship between a Pascal matrix, its inverse, and binomial inversion. It turns out that these are the same concepts viewed from two different angles. The Pascal matrix is the matrix containing Pascal’s triangle … Continue reading
Posted in binomial coefficients, matrices
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Two Methods for Proving an Alternating Binomial Identity
Recently I gave an optional homework problem in which I asked students to prove the following binomial identity: . (Here, I’m using the Iverson bracket notation in which if P is true and if P is false.) I intended for students to … Continue reading
Posted in binomial coefficients, combinatorics
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Alternating Sum of the Reciprocals of the Central Binomial Coefficients
In the last post we proved the generating function for the reciprocals of the central binomial coefficients: In this post we’re going to use this generating function to find the alternating sum of the reciprocals of the central binomial coefficients. … Continue reading
Sum of the Reciprocals of the Central Binomial Coefficients
In this post we prove the formula for the sum of the reciprocals of the central binomial coefficients : . (Of course, the sum of the central binomial coefficients themselves does not converge.) We start with the beta integral, . Replacing n … Continue reading
Posted in binomial coefficients, sequences and series
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A Probabilistic Proof of a Binomial Coefficient Identity, Generalized
In a post from a couple of years ago I gave a probabilistic proof of the binomial coefficient identity In this post I modify and generalize this proof to establish the identity As in the original proof, we use a ballsandjars … Continue reading
Posted in binomial coefficients, probability
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Generalized Binomial Coefficients from Multiplicative and Divisible Functions
Given a function f from the natural numbers to the natural numbers, one way to generalize the binomial coefficient is via . The usual binomial coefficient of course has f as the identity function . Question: What kinds of functions f guarantee that … Continue reading