
Archives
 March 2021
 December 2020
 November 2020
 October 2020
 August 2020
 July 2020
 May 2020
 March 2020
 February 2020
 January 2020
 December 2019
 September 2019
 August 2019
 July 2019
 June 2019
 May 2019
 April 2019
 March 2019
 February 2019
 January 2019
 December 2018
 November 2018
 October 2018
 September 2018
 August 2018
 July 2018
 June 2018
 May 2018
 April 2018
 March 2018
 February 2018
 January 2018
 December 2017
 November 2017
 October 2017
 September 2017
 August 2017
 July 2017
 June 2017
 May 2017
 April 2017
 March 2017
 February 2017
 January 2017
 December 2016
 November 2016
 October 2016
 September 2016
 August 2016
 July 2016
 June 2016
 May 2016
 April 2016
 March 2016
 February 2016
 January 2016
 December 2015
 November 2015
 October 2015
 September 2015
 August 2015
 July 2015
 June 2015
 May 2015
 April 2015
 March 2015
 February 2015
 January 2015
 December 2014
 November 2014
 October 2014
 September 2014
 August 2014
 July 2014
 June 2014
 May 2014
 April 2014
 March 2014
 February 2014
 January 2014
 December 2013
 November 2013
 October 2013
 September 2013
 June 2013
 May 2013
 April 2013
 March 2013
 February 2013
 January 2013
 December 2012
 November 2012
 October 2012
 September 2012
 August 2012
 July 2012
 May 2012
 March 2012
 February 2012
 January 2012
 December 2011
 November 2011
 October 2011

Meta
Category Archives: binomial coefficients
The Art of Proving Binomial Identities is Now Available
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
Posted in binomial coefficients, publishing
Leave a comment
Sum of the Reciprocals of the Binomial Coefficients
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
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
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 termbyterm. If you think about the two sides … Continue reading
Posted in binomial coefficients, probability
Leave a comment
A Sum of Ratios of Binomial Coefficients
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 SignReversing Involutions
This month’s post is on the combinatorial proof technique of signreversing 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
Posted in binomial coefficients, combinatorics
Leave a comment
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
Posted in binomial coefficients, matrices
2 Comments
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
2 Comments