A Riordan array is an infinite lower triangular matrix in which column 
has generating function 
for 
, 1, ..., where 
and 
are formal power series
over a field, 
, 
, and the coefficient
of 
in 
is nonzero (Shapiro et al. 1991). Its entry in row
and column 
is ![[x^n]g(x)f(x)^k](/images/equations/RiordanArray/Inline12.svg)
, where ![[x^n]](/images/equations/RiordanArray/Inline13.svg)
extracts the coefficient
of 
.
Writing the array as 
, ordinary matrix
multiplication gives
 |
where 
denotes function composition. The arrays
form the Riordan group, with identity
element 
and inverse element 
, where 
is the compositional inverse of 
(Shapiro et al. 1991).
For example, 
gives the infinite lower triangular Pascal matrix.
The Sprugnoli group uses a related construction
with different generating factors in even and odd columns (Barry 2026).
