Exercise 2094Combinatorics › Binomial coefficientsM1Added 3 Nov 2026

Riordan arrays (statement)

#Riordan array#generating function#binomial

Statement

Describe Riordan arrays and their use for binomial identities.

Hint

A Riordan array is defined by a pair $(g(x),f(x))$ of formal power series; entries are $[x^n]g(x)f(x)^k$ .

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Solution

Definition. The Riordan array $(g,f)$ with $g(0)\neq0$ , $f(0)=0$ , $f'(0)\neq0$ , is the infinite matrix with entries $R_{n,k}=[x^n]g(x)f(x)^k$ .

Application. Many binomial identities become matrix product identities in Riordan arrays. For instance, Pascal's triangle is the Riordan array $(1/(1-x),x/(1-x))$ .

Conclusion. Riordan arrays unify many binomial coefficient identities.

Riordan arrays (statement)

Statement

Solution

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