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 through row *m*. For example,

The inverse of a Pascal matrix turns out to be a slight variation on that Pascal matrix – one in which some of the entries are negative. For example,

More precisely, is the whose entry (starting with row 0 and column 0) is , and is the matrix whose entry (again starting with row 0 and column 0) is .

*Binomial inversion* is the following property: Given two sequences , we have

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Binomial inversion is useful for proving sum identities involving the binomial coefficients: If you can prove one such identity, you get a second identity immediately from binomial inversion.

Let’s look at the connection between Pascal matrices and binomial inversion now. Suppose you take a sequence and turn its first entries into a column vector . For example,

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Now, multiply by . This produces another vector that we can call . In other words, . For example,

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If we multiply both sides of this equation by , we get . For example,

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Let’s take a closer look at how is calculated. Entry *n*, , is the inner product of row *n* of and the vector. In other words,

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In addition, if we look at how entry *n* is calculated in via the equation, we have that is the inner product of row *n* of and the vector. In other words,

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Thus the relationship between the and sequences expressed via the Pascal matrix and its inverse is that

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This is exactly binomial inversion.

Thus binomial inversion is just expressing what the inverse of a Pascal matrix is, and knowledge of the inverse of a Pascal matrix gives you the binomial inversion formula.