The Sprugnoli group is a group of infinite lower triangular matrices with alternating column generating
functions (Barry 2026). Let 
, 
, and 
be formal power series
over a field. Require 
and require that 
and 
each have zero constant term and nonzero linear coefficient,
with 
containing only odd powers of 
. For 
, 1, ..., the generating
functions of columns 
and 
are 
and 
, respectively.
In each column, the coefficient of 
gives its entry in row 
. These matrices form a group
under ordinary matrix multiplication, with
identity matrix represented by the triple 
(Barry 2026).
A Riordan array instead uses the generating function 
for every column 
.
