Category Archives: binomial coefficients

A Request for a Proof of a Binomial Identity

Posted on March 1, 2020 by mzspivey

A few weeks ago I received an email from Professor Steve Drekic at the University of Waterloo. He asked if I knew of a way to prove the following binomial identity: (He told me later that he wanted it to … Continue reading →

Posted in binomial coefficients, generating functions | 2 Comments

Six Proofs of a Binomial Identity

Posted on August 22, 2019 by mzspivey

I’m a big fan of proving an identity in multiple ways, as I think each perspective gives additional insight into why the identity is true.  In this post we’ll work through six different proofs of the binomial identity . 1. … Continue reading →

Posted in binomial coefficients, calculus, combinatorics, generating functions, probability | 2 Comments

The Art of Proving Binomial Identities is Now Available

Posted on May 20, 2019 by mzspivey

A couple of months ago I posted that my book, The Art of Proving Binomial Identities, would soon be finished.  Well, it’s out now, and you can buy it straight from the publisher (CRC Press).  It’s up on Amazon as well. … Continue reading →

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Sum of the Reciprocals of the Binomial Coefficients

Posted on April 25, 2019 by mzspivey

In this post we’re going to prove the following identity for the sum of the reciprocals of the numbers in column k of Pascal’s triangle, valid for integers : Identity 1: The standard way to prove Identity 1 is is … Continue reading →

Posted in binomial coefficients, sequences and series | 1 Comment

The Art of Proving Binomial Identities

Posted on March 12, 2019 by mzspivey

I recently finished a book, The Art of Proving Binomial Identities, that will be published by CRC Press later this year.  We’re past the page proofs stage, and I think all that’s left on my end is to provide a little … Continue reading →

Posted in binomial coefficients, publishing | 1 Comment

Equivalence of Two Binomial Sums

Posted on October 29, 2018 by mzspivey

This month’s post entails proving the following equivalence: Identity 1: Despite the fact that these two sums are over the same range and are equal for all values of n they are not equal term-by-term. If you think about the two sides … Continue reading →

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A Sum of Ratios of Binomial Coefficients

Posted on September 26, 2018 by mzspivey

In this post we evaluate the sum .  Then we’ll generalize it and evaluate . The key tools we need for the first sum are the trinomial revision identity, , and the parallel summation identity, .  Using trinomial revision, we have . … Continue reading →

Posted in binomial coefficients | 2 Comments

An Alternating Convolution Identity via Sign-Reversing Involutions

Posted on August 1, 2018 by mzspivey

This month’s post is on the combinatorial proof technique of sign-reversing involutions.  It’s a really clever idea that can often be applied to identities that feature alternating sums.  We’ll illustrate the technique on the following identity. . (Here, we use … Continue reading →

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Determinant of a Symmetric Pascal Matrix

Posted on December 8, 2017 by mzspivey

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 →

Posted in binomial coefficients, matrices | 2 Comments

Pascal Matrices and Binomial Inversion

Posted on May 5, 2017 by mzspivey

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 | 2 Comments