Category Archives: binomial coefficients

Determinant of a Symmetric Pascal Matrix

The infinite symmetric Pascal matrix Q is given by where entry in Q is .  (Note that we begin indexing the matrix with 0, not 1, in keeping with the way Pascal’s triangle is usually indexed.) The purpose of this post is to … Continue reading

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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

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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

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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

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Generating Function for the Reciprocals of the Central Binomial Coefficients

In this post we generalize the result from the last post to find the generating function for the reciprocals of the central binomial coefficients.  As we did with that one, we start with the beta integral expression for : . Now, … Continue reading

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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

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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 balls-and-jars … Continue reading

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