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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
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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
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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
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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
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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
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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
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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
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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
Posted in binomial coefficients, matrices
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