<html><head><meta name="color-scheme" content="light dark"></head><body><pre style="word-wrap: break-word; white-space: pre-wrap;">\documentclass[12pt,reqno]{article}

\usepackage[usenames]{color}
\usepackage{amssymb}
\usepackage{amsmath}
\usepackage{amsthm}
\usepackage{amsfonts}
\usepackage{amscd}
\usepackage{graphicx}
\usepackage{mathtools}
\usepackage{mathrsfs}
\usepackage{enumerate}

\usepackage[colorlinks=true,
linkcolor=webgreen,
filecolor=webbrown,
citecolor=webgreen]{hyperref}

\definecolor{webgreen}{rgb}{0,.5,0}
\definecolor{webbrown}{rgb}{.6,0,0}

\usepackage{color}
\usepackage{fullpage}
\usepackage{float}

\usepackage{psfig}
\usepackage{graphics}
\usepackage{latexsym}
\usepackage{epsf}
\usepackage{breakurl}

\setlength{\textwidth}{6.5in}
\setlength{\oddsidemargin}{.1in}
\setlength{\evensidemargin}{.1in}
\setlength{\topmargin}{-.1in}
\setlength{\textheight}{8.4in}

\newcommand{\seqnum}[1]{\href{https://oeis.org/#1}{\underline{#1}}}

\begin{document}

\begin{center}
\epsfxsize=4in
\leavevmode\epsffile{logo129.eps}
\end{center}

\theoremstyle{plain}
\newtheorem{theorem}{Theorem}
\newtheorem{corollary}[theorem]{Corollary}
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{proposition}[theorem]{Proposition}

\theoremstyle{definition}
\newtheorem{definition}[theorem]{Definition}
\newtheorem{example}[theorem]{Example}
\newtheorem{conjecture}[theorem]{Conjecture}

\theoremstyle{remark}
\newtheorem{remark}[theorem]{Remark}

\def    \C           {\mathbb{C}}
\def    \R           {\mathbb{R}}
\def    \Q           {\mathbb{Q}}
\def    \Z           {\mathbb{Z}}
\def    \N           {\mathbb{N}}
\def    \Czero       {\mathbb{C}\backslash\{0\}}
\def    \Rzero       {\mathbb{R}\backslash\{0\}}
\def    \Qzero       {\mathbb{Q}\backslash\{0\}}
\def    \Zzero       {\mathbb{Z}\backslash\{0\}}
\def    \RzeroP      {\mathbb{R}^*}
\def    \QzeroP      {\mathbb{Q}^*}
\def    \ZzeroP      {\mathbb{Z}^*}
\def    \CzeroP      {\mathbb{C}^*}
\newcommand{\Rio}{\mathfrak{Rio}}
\newcommand{\te}{t}
\newcommand{\g}{g}
\newcommand{\f}{f}
\newcommand{\dd}{d}
\newcommand{\h}{h}
\newcommand{\id}[1]{\text{id}_{#1}}
\newcommand{\aut}[1]{\mbox{Aut}(#1)}
\newcommand{\p}[2]{\phi_{#1,#2}}
\newcommand{\prs}{\phi_{r,s}}
\newcommand{\prms}{\phi_{r,-s}}
\newcommand{\pp}[1]{\phi_{#1}}
\newcommand{\DD}{D}
\newcommand{\Dpi}{\mathcal{D}}
\newcommand{\family}{\{\prs\}_{r\in\R,s\in\R^*}}
\newcommand{\familyQ}{\{\p{a}{b}\}_{a\in\Q,b\in\QzeroP}}
\newcommand{\familyZ}{\{\p{m}{n}\}_{m\in\Z,n\in\ZzeroP}}
\newcommand{\familyR}{\{\prs(\DD)\}_{r\in\R,s\in\RzeroP}}
\newcommand{\familyQR}{\{\p{a}{b}(\DD)\}_{a\in\Q,b\in\QzeroP}}
\newcommand{\familyone}{\{\p r {\pm1}\}_{r\in\R}}
\newcommand{\familyminus}{\{\p r {-1}\}_{r\in\R}}
\newcommand{\familyRio}{\{\prs(\Rio)\}_{r\in\R,s\in\RzeroP}}
\newcommand{\ene}{n}
\newcommand{\familyS}{(\pp\ene(\DD))_{n\geq1}}
\newcommand{\familySm}{(\pp{-\ene}(\DD))_{n\geq1}}
\newcommand{\familySf}{(\pp{1/\ene}(\DD))_{n\geq1}}
\newcommand{\familySfm}{(\pp{-1/\ene}(\DD))_{n\geq1}}
\newcommand{\mis}[2]{\mathfrak{Col}_{#1}^{#2}}
\newcommand{\misSum}[1]{\mathfrak{RowS}^{#1}}
\newcommand{\misD}[2]{\mathfrak{Diag}_{#1}^{#2}}
\newcommand{\rowsum}[2]{\mathfrak{r}_{#1}^{#2}}
\newcommand{\TT}{T}
\newcommand{\Tc}{\mathfrak{row}^P}
\newcommand{\misc}[2]{\mathfrak{col}_{#1}^{#2}}
\newcommand{\missum}[1]{\mathfrak{rowS}^{#1}}
\newcommand{\misd}[2]{\mathfrak{diag}_{#1}^{#2}}
\newcommand{\SSs}[1]{\mathfrak{D}^{#1}}
\newcommand{\ZZ}[1]{\mathfrak{Z}^{#1}}
\newcommand{\zz}[1]{\mathfrak{z}^{#1}}
\newcommand{\BB}[1]{\mathfrak{B}^{#1}}
\newcommand{\bb}[1]{\mathfrak{b}^{#1}}
\newcommand{\Bmis}[2]{\mathfrak{BCol}_{#1}^{#2}}
\newcommand{\BmisSum}[1]{\mathfrak{BRowS}^{#1}}
\newcommand{\BmisD}[2]{\mathfrak{BDiag}_{#1}^{#2}}



\begin{center}
\vskip 1cm
{\LARGE\bf Square Matrices Generated by \\
\vskip .1in
Sequences of Riordan Arrays}
\vskip 1cm
\large
\^Angela Mestre and Jos\'e Agapito\\
Centro de An\'alise Funcional, Estruturas Lineares e Aplica\c{c}\~oes\\
Grupo de Estruturas Alg\'ebricas, Lineares e Combinat\'orias\\
Departamento de Matem\'atica\\
Faculdade de Ci\^encias, Universidade de Lisboa\\
1749-016 Lisboa\\
Portugal\\
\href{mailto:amestre@fc.ul.pt}{\tt amestre@fc.ul.pt} \\
\href{mailto:jaruiz@fc.ul.pt}{\tt jaruiz@fc.ul.pt} \\
\end{center}

\vskip .2in

\begin{abstract}
We consider sequences of images of Riordan arrays 
under some Riordan group 
automorphisms introduced by Bacher. We enclose their properties 
into several infinite square arrays which turn out to be 
of  combinatorial interest.  To illustrate our approach we consider Cameron and 
Nkwanta's sequence 
of generalized RNA arrays (whose first term is the well-known Nkwanta RNA 
array). Although this sequence of generalized RNA arrays was originally 
established 
as a sequence of pseudo-involutions, 
we show that it does not contain pseudo-involutions other than Nkwanta's array.
We also show that these arrays 
are actually images of a new array under some of Bacher's automorphisms. We 
study the combinatorics of some square matrices related to the generalized RNA 
arrays and to sequences of genuine pseudo-involutions  
generated by Nkwanta's array.
\end{abstract}
\section{Introduction}
\label{sec:intro}
Let 
$\dd(t)=1+\sum_{k=1}^\infty d_kt^k$
and
$\h(t)=\sum_{k=1}^\infty h_kt^k$
be two generating functions over the field of complex numbers 
$\C$
with
$h_1\neq0$. 
A {\em Riordan array} is an infinite lower triangular matrix 
$\DD=(\DD_{k,n})_{k\geq n\geq 1}$, 
usually represented as a pair 
$\DD=(\dd(\te),\h(\te))$, 
whose 
entries are given by 
\begin{equation}
\label{eq:entries}
\DD_{k,n}=[\te^{k-1}]\dd(\te)\h(\te)^{n-1} ,
\end{equation}
where   
$[\te^k]$
is the usual coefficient extraction operator. The set of Riordan arrays 
forms a group formally introduced by Shapiro, Getu, Woan, and Woodson 
\cite{ShGeWoWo:Rgroup}, 
here denoted by 
$\Rio$.
The group multiplication is given by
\begin{equation}
\label{eq:multR}
(\dd_{1}(\te),\h_{1}(\te)){}
(\dd_{2}(\te),\h_{2}(\te))=(\dd_{1}(\te)\dd_{2}(\h_{1}(\te)),\h_{2}(\h_{1}
(\te))) ,
\end{equation}
while the group identity is 
$\id {\Rio}=(1,\te)$. 
The 
inverse of any element 
$\DD=(\dd(\te),\h(\te))$ 
is 
$\DD^{-1}=(1/\dd(\bar{\h}(\te)),\bar{\h}(\te))$,
where 
$\bar{\h}(\te)$
denotes the compositional inverse of the generating function 
$\h(t)$, 
i.e., 
$\h(\bar{\h}(t))=\bar{\h}(\h(t))=t$.
There exists an extensive literature on the Riordan group and related concepts 
as, for instance,
\cite{pw,bijections,cameron46,HHS:RioGvsSheffG,CKS:inv,HeSprug:seqChar,cheon,
hitt,jean-louis,LMM}.
Throughout the years, Riordan arrays have been revealed to be very powerful 
tools in combinatorics, see, e.g.,
\cite{sprugnoli,MSRV,merlini-verri,LM,idInduced,HeSha:sum}.

This paper introduces several square arrays associated with sequences of 
Riordan arrays. The main intention is to encompass some collective properties 
of the arrays in the sequence into a properly constructed square array. Let 
$(\DD_n)_{n\geq1}$
be a sequence of Riordan arrays. Fix integers 
$i,j\geq1$.
From each of the arrays in the sequence take the 
$i$th column 
(resp., the $j$th diagonal or the row-sum vector). 
Build a square matrix such that the 
$n$th 
column corresponds to the 
$i$th column 
(resp., the $j$th diagonal or the row-sum vector) 
of the matrix 
$\DD_n$. 
Here, the Riordan array sequences are generated by the 
images of a given Riordan element under  a sequence of Riordan group 
automorphisms. We shall refer to the Riordan group element that generates the 
matrix sequence, as the 
{\em base array} 
of the sequence.

Given
$\alpha,\gamma\in\C$ 
and 
$\beta\in \CzeroP$, 
with
$\CzeroP=\Czero$,  
Bacher 
\cite{bacher} 
introduced the automorphisms 
$\varphi_{\alpha,\beta,\gamma}:\Rio\to\Rio$ 
as follows
\begin{eqnarray}
\nonumber
\varphi_{\alpha,\beta,\gamma}:\Rio &amp; \to &amp; \Rio\\
\label{eq:bacher}
(\dd(\te),\h(\te)) &amp; \mapsto &amp; \left(\biggl(\frac{\h(\te)}{\te}\biggr)^\alpha
\dd(\te)^\beta\h'(\te)^\gamma,\h(\te)\right) ,
\end{eqnarray}
where 
$\h'(\te)$
denotes the derivative of 
$\h(\te)$.
In particular, the automorphism 
$\varphi_{1,1,0}$
is usually known as the 
{\em diagonal translation operator} 
\cite{bacher,complementary}, 
for 
$\varphi_{1,1,0}(\DD)$ 
is the matrix obtained from 
$\DD$ 
by removing the first column and the first row 
(recall formula \eqref{eq:entries}).

This paper is organized as follows. Given a Riordan array 
$\DD$ 
and fixed complex numbers 
$\beta\in\CzeroP$ 
and 
$\gamma\in\C$, 
Section 
\ref{sec:matrices} 
defines infinite square matrices whose columns are obtained by extracting a 
column, a diagonal slice, or the row sum sequence of each array in the 
sequences 
$(\varphi_{1,n,\gamma}(\DD))_{n\geq 1}$ 
and 
$(\varphi_{1,\beta,l}(\DD))_{l\geq 1}$. 
We give examples which show that these new matrices are compelling 
combinatorial devices enclosing the joint properties of the arrays in the 
sequences. 

Section 
\ref{sec:sequences} 
expresses the 
$A$, 
$Z$-sequences 
\cite{MSRV,HeSprug:seqChar} 
and the 
$B(x)$-sequence 
\cite{yang} 
of the image of a Riordan array 
$\DD$ 
under an automorphism 
$\varphi_{\alpha,\beta,0}$
in terms of the corresponding sequences of the base array. We define again 
infinite square matrices associated with the 
$Z$ 
and 
$B(x)$-sequences 
of the arrays in the sequence 
$(\varphi_{1,n,\gamma}(\DD))_{n\geq 1}$. 
We illustrate our setting with the classical Pascal array.

Section 
\ref{sec:inv} 
deals with pseudo-involutions of the Riordan group under the action of Bacher's 
automorphisms. We prove that they map pseudo-involutions to pseudo-involutions. 
We also show that the generalized RNA arrays of Cameron and Nkwanta 
\cite{cameron46} 
form an array sequence 
$(\varphi_{1, n,0}(\Dpi))_{n\geq 1}$, 
where 
$\Dpi$ 
is a new Riordan array. We show that contrary to what has been believed over 
the years, the arrays in this sequence are not pseudo-involutions.
We define instead sequences of authentic pseudo-involutions  by applying 
Bacher's automorphisms to Nkwanta's RNA triangle. We determine some of the 
infinite matrices of Section 
\ref{sec:matrices} 
for both the generalized RNA arrays and the new sequences of pseudo-involutions 
having Nkwanta's array as base. 

Along the paper, all the closed forms for sequences that we give and the 
correspondence to sequences in the On-line Encyclopedia of Integer Sequences 
\cite{oeis} 
were verified using Mathematica up to order 
$n=100$.
Note that often our sequences match the assigned sequence numbers only from a 
given element on.
\section{Infinite matrices via Bacher's automorphisms with 
nonnegative parameters}
\label{sec:infinite}
This section introduces several infinite square matrices associated with 
Riordan array sequences through Bacher's automorphisms 
parametrized by 
$(1,n,\gamma)$ 
or 
$(1,\beta,l)$ 
with 
$n,l\geq1$.

Let 
$\DD=(\dd(\te),\h(\te))$ 
denote a Riordan array. We restrict the parameter 
$\alpha$
in the definition of Bacher's automorphisms 
\eqref{eq:bacher} 
to nonnegative integers. Then, for fixed 
$\beta\in\CzeroP$
and 
$\gamma\in\C$
and
nonnegative integers 
$m'\geq m$, 
we have 
$
\varphi_{m',\beta,\gamma}(\DD)=
(\varphi_{m'-m,1,0}\circ\varphi_{m,\beta,\gamma})(\DD)
$.  
Thus,  from a combinatorial perspective it is enough to examine what 
happens for 
$m=1$, 
which we adopt in the sequel. To shorten notation, we define
$$
\phi_{\beta,\gamma}=\varphi_{1,\beta,\gamma} 
\quad
\text{and}
\quad
\pp n=\varphi_{1,n,0} .
$$
Also, as to the notation, note that throughout the paper we shall omit the 
index 0 associated with 
$\gamma=0$. 
\subsection{The array sequence $(\phi_{n,\gamma}(\DD))_{n\geq1}$}
\label{sec:matrices}
Among the sequences that can be associated with a Riordan 
array 
$\DD$, 
we focus on the columns, diagonal slices, and the row sum sequence.
We construct infinite matrices by putting together columns, diagonal slices, or 
row sum sequences, extracted from each Riordan array in the sequence 
$(\phi_{\ene,\gamma}(\DD))_{n\geq 1}$.
\begin{definition}
\label{def:infinite}
Fix a number $\gamma\in\C$.
For any Riordan array  
$\DD=(\dd(t),h(t))$ 
and any integer 
$i\geq 1$, 
define the following infinite matrices:
\begin{enumerate}[(i)]
\item 
Matrix $\mis i {\gamma,\DD}$, 
whose 
$\ene$th 
column is the 
$i$th 
column of 
$\phi_{\ene,\gamma}(\DD)$:
\begin{equation}
\label{eq:col}
(\mis i  
{\gamma,\DD})_{k,\ene}=(\phi_{\ene,\gamma}(\DD))_{k,i}
                      =[\te^{k+i-1}]\dd(\te)^\ene\h'(t)^\gamma\h(\te)^{i} , 
\quad 
k,\ene\geq1 .
\end{equation}
\item 
Matrix  
$\misD i {\gamma,\DD}$, 
whose 
$\ene$th 
column is the 
$i$th 
(descending)
diagonal slice of  
$\phi_{\ene,\gamma}(\DD)$:
\begin{equation}
\label{eq:diag}
(\misD i {\gamma,\DD})_{k,\ene}=
(\phi_{\ene,\gamma}(\DD))_{k+i,k}=
[\te^{k+i-1}]\dd(\te)^\ene h'(t)^\gamma\h(\te)^k , 
\quad 
k,\ene\geq1 .
\end{equation}
\item 
Let
$(\rowsum {\ene}{\gamma,\DD})_k$, 
$k\geq 1$, 
be the sequence whose 
$k$th 
term is the sum of the entries of the 
$k$th 
row of  
$\phi_{\ene,\gamma}(\DD)$, 
for 
$n\geq1$. 
In this context, we define the matrix 
$\misSum  {\gamma,\DD}$,
whose 
$\ene$th 
column is the sequence of the row sums of 
$\phi_{\ene,\gamma}(\DD)$:
\begin{equation}
\label{eq:row}
(\misSum {\gamma,\DD})_{k,\ene}=
(\rowsum {\ene}{\gamma,\DD})_{k}=
[t^{k}]\frac{h(t)d(t)^nh'(t)^\gamma}{(1-h(t))} , 
\quad 
k,\ene\geq1 .
\end{equation}
\end{enumerate}
\end{definition}
In the last equation, we used the fact that the 
generating function for the row sum of a Riordan array 
$(d(t),h(t))$ 
is given by 
$\dd(\te)/(1-\h(\te))$ 
\cite{bijections,HeSha:sum}.

Note that analogous matrix definitions hold for the sequences  
$(\phi_{-\ene,\gamma}(\DD))_{n\geq1}$. 
In this context, the 
$\ene$th column 
of the similar matrices, denoted 
$\misc i {\gamma,\DD}$, $\misd i {\gamma,\DD}$, 
and 
$\missum{\gamma,\DD}$, 
corresponds to the 
$i$th 
column, the 
$i$th 
diagonal slice, or the row sum sequence of the matrix 
$\phi_{-\ene,\gamma}(\DD)$. 
Actually, given a Riordan array 
$\DD$, 
we may even extend the definitions to the sequences    
$\phi_{\pm1/n,\gamma}(\DD))_{n\geq1}$ 
so that the 
$n$th 
column of square  matrices is associated with 
a column, diagonal slice or the row sum sequence of the matrix 
$\phi_{\pm1/n,\gamma}(\DD)$.

Moreover, to avoid zero rows, in the  definition of the square array 
$\mis i{\gamma,\DD}$, 
we removed the first 
$i-1$
zero elements of the 
$i$th 
column of the matrices 
$\phi_{n,\gamma} (\DD)$ 
for all 
$i,n\geq 1$. 
When 
$\gamma=0$ 
this definition may be seen as  reciprocal
to that of Riordan arrays, for the power of the second  element of 
the pair 
$(\dd(\te),\h(\te))$ 
remains fixed, while the power of the first element increases as the number of 
columns increases.  Indeed, one may even transform the matrix 
$\mis i \DD$ 
into a regular Riordan array by multiplying the entry 
$(k,n)$  
by 
$t^{n-1}$. 
The resulting array is defined in terms of the automorphism 
$\pp i$
as given in the next remark.
\begin{remark}
\label{rem:col}
Given any Riordan array 
$\DD=(\dd(t),h(t))$, 
let 
$\DD_{S}=(h(t)/t,td(t))$. 
Then, for any 
$i,n\geq1$,   
the 
$n$th 
column of the following Riordan array 
\begin{equation*} 
\pp i(\DD_{S})=
\left(
\left(\frac{h(t)}{t}\right)^id(t),td(t)
\right)
\quad
\text{with}
\quad
(\pp i(\DD_S))_{k,n}=
[\te^{k+i-n}]\dd(\te)^{\ene}\h(\te)^{i} , 
\quad 
k,\ene\geq1 ,
\end{equation*} 
from the 
$n$th 
row on, equals the 
$n$th 
column of the square array 
$\mis i {\DD}$  
(formula \eqref{eq:col}).
\end{remark}
Thus, both 
$(\pp n(\DD))_{n\geq1}$ 
and 
$(\pp {n}(\DD_S))_{n\geq1}$ 
carry the same combinatorial information regarding the columns.
\begin{proposition}
Let 
$\DD=(\dd(t),h(t))$ 
be any Riordan array and let 
$\DD_{S}=(h(t)/t,td(t))$. 
Then, there is a 1-1 correspondence between the columns of the matrices in the 
array sequence 
$(\pp n(\DD))_{n\geq1}$ 
and those in the sequence 
$(\pp {n}(\DD_S))_{n\geq1}$
such that the 
$i$th 
column of 
$\pp n(\DD)$ 
is associated with the 
$n$th 
column of 
$\pp i(\DD_S)$ 
for any 
$i,n\geq1$.
\end{proposition}
Furthermore, some of the next examples show that the matrices of Definition 
\ref{def:infinite}  
highlight the shared  properties of the matrices in the sequence 
$(\pp \ene (\DD))_{n\geq1}$.
\begin{example}
\label{ex:pascal1}
\normalfont
We study the array sequence 
$(\pp{-\ene}(P))_{n\geq1}$,
where 
$P=(1/(1-\te),\te/(1-\te))$ 
is the classical Pascal array:
\begin{center}
\begin{equation}
\label{eq:pascal}
P=
\left[
\begin{array}{rrrrrrrrrrrrrr}
1     &amp; 0    &amp; 0    &amp; 0    &amp; 0    &amp; 0    &amp; 0    &amp;\cdots\\ 
1     &amp; 1    &amp; 0    &amp; 0    &amp; 0    &amp; 0    &amp; 0    &amp;\cdots\\ 
1     &amp; 2    &amp; 1    &amp; 0    &amp; 0    &amp; 0    &amp; 0    &amp;\cdots\\ 
1     &amp; 3    &amp; 3    &amp; 1    &amp; 0    &amp; 0    &amp; 0    &amp;\cdots\\ 
1     &amp; 4    &amp; 6    &amp; 4    &amp; 1    &amp; 0    &amp; 0    &amp;\cdots\\ 
1     &amp; 5    &amp; 10   &amp; 10   &amp; 5    &amp; 1    &amp; 0    &amp;\cdots\\ 
1     &amp; 6    &amp; 15   &amp; 20   &amp; 15   &amp; 6    &amp; 1    &amp;\cdots\\ 
\vdots&amp;\vdots&amp;\vdots&amp;\vdots&amp;\vdots&amp;\vdots&amp;\vdots&amp;\ddots  
\end{array}
\right] .
\end{equation}
\end{center}
The first arrays 
$\pp{-\ene} (P)$ 
are
$$
\pp {-1}(P)=
\left[
\begin{array}{llllllll}
1     &amp; 0    &amp; 0    &amp; 0    &amp; 0    &amp; 0    &amp;0     &amp;\cdots\\ 
0     &amp; 1    &amp; 0    &amp; 0    &amp; 0    &amp; 0    &amp;0     &amp;\cdots\\ 
0     &amp; 1    &amp; 1    &amp; 0    &amp; 0    &amp; 0    &amp;0     &amp;\cdots\\ 
0     &amp; 1    &amp; 2    &amp; 1    &amp; 0    &amp; 0    &amp;0     &amp;\cdots\\ 
0     &amp; 1    &amp; 3    &amp; 3    &amp; 1    &amp; 0    &amp;0     &amp;\cdots\\ 
0     &amp; 1    &amp; 4    &amp; 6    &amp; 4    &amp; 1    &amp;0     &amp;\cdots\\ 
0     &amp; 1    &amp; 5    &amp; 10   &amp; 10   &amp; 5    &amp;1     &amp;\cdots\\ 
\vdots&amp;\vdots&amp;\vdots&amp;\vdots&amp;\vdots&amp;\vdots&amp;\vdots&amp;\ddots  
\end{array}
\right] , 
$$
$$
\pp {-2}(P)=
\left[
\begin{array}{rrrrrrrrr}
1     &amp;0     &amp; 0    &amp; 0    &amp; 0    &amp; 0    &amp; 0    &amp;0     &amp;\cdots\\
-1    &amp;1     &amp; 0    &amp; 0    &amp; 0    &amp; 0    &amp; 0    &amp;0     &amp;\cdots\\ 
0     &amp;0     &amp; 1    &amp; 0    &amp; 0    &amp; 0    &amp; 0    &amp;0     &amp;\cdots\\ 
0     &amp;0     &amp; 1    &amp; 1    &amp; 0    &amp; 0    &amp; 0    &amp;0     &amp;\cdots\\ 
0     &amp;0     &amp; 1    &amp; 2    &amp; 1    &amp; 0    &amp; 0    &amp;0     &amp;\cdots\\ 
0     &amp;0     &amp; 1    &amp; 3    &amp; 3    &amp; 1    &amp; 0    &amp;0     &amp;\cdots\\ 
0     &amp;0     &amp; 1    &amp; 4    &amp; 6    &amp; 4    &amp; 1    &amp;0     &amp;\cdots\\ 
0     &amp;0     &amp; 1    &amp; 5    &amp; 10   &amp; 10   &amp; 5    &amp;1     &amp;\cdots\\ 
\vdots&amp;\vdots&amp;\vdots&amp;\vdots&amp;\vdots&amp;\vdots&amp;\vdots&amp;\vdots&amp;\ddots  
\end{array}
\right] ,
$$
$$
\pp {-3}(P)=
\left[
\begin{array}{rrrrrrrrrrr}
1     &amp; 0    &amp;0     &amp; 0    &amp; 0    &amp; 0    &amp; 0    &amp; 0    &amp; 0    &amp;\cdots\\
-2    &amp; 1    &amp;0     &amp; 0    &amp; 0    &amp; 0    &amp; 0    &amp; 0    &amp; 0    &amp;\cdots\\
1     &amp; -1   &amp;1     &amp; 0    &amp; 0    &amp; 0    &amp; 0    &amp; 0    &amp; 0    &amp;\cdots\\ 
0     &amp; 0    &amp;0     &amp; 1    &amp; 0    &amp; 0    &amp; 0    &amp; 0    &amp; 0    &amp;\cdots\\ 
0     &amp; 0    &amp;0     &amp; 1    &amp; 1    &amp; 0    &amp; 0    &amp; 0    &amp; 0    &amp;\cdots\\ 
0     &amp; 0    &amp;0     &amp; 1    &amp; 2    &amp; 1    &amp; 0    &amp; 0    &amp; 0    &amp;\cdots\\ 
0     &amp; 0    &amp;0     &amp; 1    &amp; 3    &amp; 3    &amp; 1    &amp; 0    &amp; 0    &amp;\cdots\\ 
0     &amp; 0    &amp;0     &amp; 1    &amp; 4    &amp; 6    &amp; 4    &amp; 1    &amp; 0    &amp;\cdots\\ 
0     &amp; 0    &amp;0     &amp; 1    &amp; 5    &amp; 10   &amp; 10   &amp; 5    &amp; 1    &amp;\cdots\\ 
\vdots&amp;\vdots&amp;\vdots&amp;\vdots&amp;\vdots&amp;\vdots&amp;\vdots&amp;\vdots&amp;\vdots&amp;\ddots
\end{array}
\right] ,
$$
$$
\pp{-4}(P)=
\left[ 
\begin{array}{rrrrrrrrrrr}
1     &amp;0     &amp;0     &amp; 0    &amp; 0    &amp; 0    &amp; 0    &amp; 0    &amp; 0    &amp;0     &amp;\cdots\\
-3    &amp;1     &amp;0     &amp; 0    &amp; 0    &amp; 0    &amp; 0    &amp; 0    &amp; 0    &amp;0     &amp;\cdots\\
3     &amp;-2    &amp;1     &amp; 0    &amp; 0    &amp; 0    &amp; 0    &amp; 0    &amp; 0    &amp;0     &amp;\cdots\\
-1    &amp;1     &amp;-1    &amp; 1    &amp; 0    &amp; 0    &amp; 0    &amp; 0    &amp; 0    &amp;0     &amp;\cdots\\
0     &amp;0     &amp;0     &amp; 0    &amp; 1    &amp; 0    &amp; 0    &amp; 0    &amp; 0    &amp;0     &amp;\cdots\\ 
0     &amp;0     &amp;0     &amp; 0    &amp; 1    &amp; 1    &amp; 0    &amp; 0    &amp; 0    &amp;0     &amp;\cdots\\ 
0     &amp;0     &amp;0     &amp; 0    &amp; 1    &amp; 2    &amp; 1    &amp; 0    &amp; 0    &amp;0     &amp;\cdots\\ 
0     &amp;0     &amp;0     &amp; 0    &amp; 1    &amp; 3    &amp; 3    &amp; 1    &amp; 0    &amp;0     &amp;\cdots\\ 
0     &amp;0     &amp;0     &amp; 0    &amp; 1    &amp; 4    &amp; 6    &amp; 4    &amp; 1    &amp;0     &amp;\cdots\\ 
0     &amp;0     &amp;0     &amp; 0    &amp; 1    &amp; 5    &amp; 10   &amp; 10   &amp; 5    &amp;1     &amp;\cdots\\ 
\vdots&amp;\vdots&amp;\vdots&amp;\vdots&amp;\vdots&amp;\vdots&amp;\vdots&amp;\vdots&amp;\vdots&amp;\vdots&amp;\ddots 
\end{array}
\right] .
$$
Clearly, from the 
$(n+1)$th
column on we recover Pascal's array. We now consider the row sum matrix 
$\missum P$
associated with the sequence 
$(\pp{-\ene}(P))_{n\geq1}$,
whose 
$n$th 
column yields the row sum sequence of the matrix 
$\pp{-n}(P)$:
$$
\missum{P}=
\left[
\begin{array}{rrrrrrrrr}
1     &amp; 1    &amp; 1    &amp; 1    &amp; 1    &amp; 1    &amp;  1   &amp; \cdots\\ 
1     &amp; 0    &amp;-1    &amp; -2   &amp; -3   &amp; -4   &amp; -5   &amp;\cdots\\ 
2     &amp; 1    &amp; 1    &amp; 2    &amp; 4    &amp; 7    &amp;  11  &amp;\cdots\\ 
4     &amp; 2    &amp; 1    &amp; 0    &amp; -2   &amp; -6   &amp; -13  &amp;\cdots\\
8     &amp; 4    &amp; 2    &amp; 1    &amp; 1    &amp; 3    &amp;  9   &amp;\cdots\\ 
16    &amp; 8    &amp; 4    &amp; 2    &amp; 1    &amp; 0    &amp; -3   &amp;\cdots\\ 
32    &amp; 16   &amp; 8    &amp; 4    &amp; 2    &amp; 1    &amp;  1   &amp;\cdots\\
\vdots&amp;\vdots&amp;\vdots&amp;\vdots&amp;\vdots&amp;\vdots&amp;\vdots&amp;\ddots
\end{array}
\right] .
$$
We have 
$(\missum{P})_{k,n}= \sum_{i=0}^{k-1}(-1)^i{n-k+i\choose i}$.
Note that from the 
$(n+1)$th 
element on, the 
$n$th 
column is the row sum sequence of the Pascal array 
\seqnum{A000079} 
whose elements are powers of 2. Now, we examine the upper triangular matrix  
$\Tc$ 
obtained by considering only the 
$n$ 
first elements of the 
$n$th 
column of 
$\missum P$ 
for each 
$n\geq1$:
\begin{center}
$
\Tc=
\left[
\begin{array}{rrrrrrrrr}
1     &amp; 1    &amp; 1    &amp; 1    &amp; 1    &amp; 1    &amp;  1   &amp;\cdots\\ 
0     &amp; 0    &amp; -1   &amp; -2   &amp; -3   &amp; -4   &amp;  -5  &amp;\cdots\\ 
0     &amp; 0    &amp; 1    &amp; 2    &amp; 4    &amp; 7    &amp;  11  &amp;\cdots\\ 
0     &amp; 0    &amp; 0    &amp; 0    &amp; -2   &amp; -6   &amp; -13  &amp;\cdots\\
0     &amp; 0    &amp; 0    &amp; 0    &amp; 1    &amp; 3    &amp;  9   &amp;\cdots\\ 
0     &amp; 0    &amp; 0    &amp; 0    &amp; 0    &amp; 0    &amp; -3   &amp;\cdots\\ 
0     &amp; 0    &amp; 0    &amp; 0    &amp; 0    &amp; 0    &amp;  1   &amp;\cdots\\
\vdots&amp;\vdots&amp;\vdots&amp;\vdots&amp;\vdots&amp;\vdots&amp;\vdots&amp;\ddots
\end{array}
\right] .
$
\end{center}
Then, the matrix 
$\Tc$ 
is the sequence 
\seqnum{A220074} 
as an upper right triangle.  
\end{example}
We now study the array sequences 
$(\pp n (\DD))_{n\geq1}$, 
where 
$\DD$ 
denotes the triangles  
\seqnum{A025581} 
and 
\seqnum{A114284}.
\begin{example}
\label{ex:A025581}
\normalfont
We consider the Appell group element 
$\TT=(p(\te)^2,\te)=(1/(1-\te)^2,\te)$ 
which is the triangle 
$\TT'=(\te p(\te)^2,t)$ 
\seqnum{A025581} 
without the first row. Note that 
$\TT'$ 
is not in the Riordan group, for the entry (1,1) is zero. The columns of 
$\TT$ 
yield the second column of Pascal's triangle 
$P=(p(\te), tp(t))$.
\begin{center}
$
\TT=
\left[
\begin{array}{llllllll}
1     &amp; 0    &amp; 0    &amp; 0    &amp; 0    &amp; 0    &amp;\cdots\\ 
2     &amp; 1    &amp; 0    &amp; 0    &amp; 0    &amp; 0    &amp;\cdots\\ 
3     &amp; 2    &amp; 1    &amp; 0    &amp; 0    &amp; 0    &amp;\cdots\\ 
4     &amp; 3    &amp; 2    &amp; 1    &amp; 0    &amp; 0    &amp;\cdots\\ 
5     &amp; 4    &amp; 3    &amp; 2    &amp; 1    &amp; 0    &amp;\cdots\\ 
6     &amp; 5    &amp; 4    &amp; 3    &amp; 2    &amp; 1    &amp;\cdots\\ 
\vdots&amp;\vdots&amp;\vdots&amp;\vdots&amp;\vdots&amp;\vdots&amp;\ddots  
\end{array}
\right] .
$
\end{center}
Likewise, the images of 
$\TT$ 
under the automorphisms 
$\pp n$ 
yield the 
$(2n)$th 
columns of the Pascal array 
$P$.
Thus, the columns of the matrices 
$\mis i {\TT}$ 
(see formula \eqref{eq:col}) 
are the even columns of Pascal's triangle for any 
$i\geq 1$:
\begin{center}
$
\mis i {\TT}=
\left[
\begin{array}{llllllll}
1     &amp; 1    &amp; 1    &amp; 1    &amp; 1    &amp; 1    &amp;\cdots\\ 
2     &amp; 4    &amp; 6    &amp; 8    &amp; 10   &amp; 12   &amp;\cdots\\ 
3     &amp; 10   &amp; 21   &amp; 36   &amp; 55   &amp; 78   &amp;\cdots\\ 
4     &amp; 20   &amp; 56   &amp; 120  &amp; 220  &amp; 364  &amp;\cdots\\
5     &amp; 35   &amp; 126  &amp; 330  &amp; 715  &amp; 1365 &amp;\cdots\\ 
6     &amp; 56   &amp; 252  &amp; 792  &amp; 2002 &amp; 4368 &amp;\cdots\\ 
7     &amp; 84   &amp; 462  &amp; 1716 &amp; 5005 &amp; 12376&amp;\cdots\\
\vdots&amp;\vdots&amp;\vdots&amp;\vdots&amp;\vdots&amp;\vdots&amp;\ddots
\end{array}
\right] .
$
\end{center}
Indeed, 
$(\mis i {\TT})_{k,n}={2n+k-2\choose k-1}$, 
$k,n\geq 1$. 
The
first sequence numbers are 
\seqnum{A000027},
\seqnum{A000292}, 
\seqnum{A000389}, 
\seqnum{A000580}, 
\seqnum{A000582}, 
\seqnum{A001288}, 
\seqnum{A010966}, 
and 
\seqnum{A010968}. 
From the second row on, the first sequence numbers are 
\seqnum{A005843}, 
\seqnum{A014105}, 
\seqnum{A002492}, 
\seqnum{A053126}, 
\seqnum{A053127}, 
and 
\seqnum{A053128}.
\end{example}
\begin{example}
\label{ex:A114284}
\normalfont
We consider the array sequence 
$(\phi_n(A))_{n\geq 1}$, 
where 
$A=((1-3\te)/(1-\te),\te)$
\seqnum{A114284}: 
\begin{center}
$
A=
\left[
\begin{array}{rrrrrrrrrrr}
1     &amp; 0    &amp;0     &amp;0     &amp; 0    &amp;0     &amp;\cdots\\ 
-2    &amp; 1    &amp;0     &amp;0     &amp; 0    &amp;0     &amp;\cdots\\ 
-2    &amp; -2   &amp;1     &amp;0     &amp; 0    &amp;0     &amp;\cdots\\ 
-2    &amp; -2   &amp;-2    &amp;1     &amp; 0    &amp;0     &amp;\cdots\\ 
-2    &amp; -2   &amp;-2    &amp;-2    &amp; 1    &amp;0     &amp;\cdots\\
-2    &amp; -2   &amp;-2    &amp; -2   &amp; -2   &amp;1     &amp;\cdots\\ 
\vdots&amp;\vdots&amp;\vdots&amp;\vdots&amp;\vdots&amp;\vdots&amp;\ddots
\end{array}
\right] .
$
\end{center}
The base array 
$A$ 
coincides with 
$\pp 1 (A)$. 
The next arrays 
$\pp n (A)$ 
are
$$
\pp 2 (A)=
\left[ 
\begin{array}{rrrrrrrrrr}
1     &amp; 0    &amp; 0    &amp; 0    &amp; 0    &amp; 0    &amp;\cdots\\ 
-4    &amp; 1    &amp; 0    &amp; 0    &amp; 0    &amp; 0    &amp;\cdots\\ 
0     &amp; -4   &amp; 1    &amp; 0    &amp; 0    &amp; 0    &amp;\cdots\\ 
4     &amp; 0    &amp; -4   &amp; 1    &amp; 0    &amp; 0    &amp;\cdots\\ 
8     &amp; 4    &amp; 0    &amp; -4   &amp; 1    &amp; 0    &amp;\cdots\\ 
12    &amp; 8    &amp; 4    &amp; 0    &amp; -4   &amp; 1    &amp;\cdots\\ 
\vdots&amp;\vdots&amp;\vdots&amp;\vdots&amp;\vdots&amp;\vdots&amp;\ddots
\end{array}
\right] ,
$$
$$
\pp 3(A)=
\left[
\begin{array}{rrrrrrrrrrr}
1     &amp; 0    &amp; 0    &amp; 0    &amp; 0    &amp;0     &amp;\cdots\\ 
-6    &amp; 1    &amp; 0    &amp; 0    &amp; 0    &amp;0     &amp;\cdots\\
6     &amp; -6   &amp; 1    &amp; 0    &amp; 0    &amp;0     &amp;\cdots\\
10    &amp; 6    &amp; -6   &amp; 1    &amp; 0    &amp;0     &amp;\cdots\\ 
6     &amp; 10   &amp; 6    &amp; -6   &amp; 1    &amp;0     &amp;\cdots\\ 
-6    &amp; 6    &amp; 10   &amp; 6    &amp; -6   &amp;1     &amp;\cdots\\ 
\vdots&amp;\vdots&amp;\vdots&amp;\vdots&amp;\vdots&amp;\vdots&amp;\ddots
\end{array}
\right] ,
$$
$$
\pp 4 (A)=
\left[ 
\begin{array}{rrrrrrrrrrr}
1     &amp; 0    &amp; 0    &amp; 0    &amp; 0    &amp;0     &amp;\cdots\\ 
-8    &amp; 1    &amp; 0    &amp; 0    &amp; 0    &amp;0     &amp;\cdots\\ 
16    &amp; -8   &amp; 1    &amp; 0    &amp; 0    &amp;0     &amp;\cdots\\ 
8     &amp; 16   &amp; -8   &amp; 1    &amp; 0    &amp;0     &amp;\cdots\\ 
-16   &amp; 8    &amp; 16   &amp; -8   &amp; 1    &amp;0     &amp;\cdots\\ 
-40   &amp; -16  &amp; 8    &amp; 16   &amp; -8   &amp;1     &amp;\cdots\\ 
\vdots&amp;\vdots&amp;\vdots&amp;\vdots&amp;\vdots&amp;\vdots&amp;\ddots
\end{array}
\right] ,
$$
$$
\pp 5 (A)=
\left[
\begin{array}{rrrrrrrrrrr}
1     &amp; 0    &amp; 0    &amp; 0    &amp; 0    &amp;0     &amp;\cdots\\ 
-10   &amp; 1    &amp; 0    &amp; 0    &amp; 0    &amp;0     &amp;\cdots\\ 
30    &amp; -10  &amp; 1    &amp; 0    &amp; 0    &amp;0     &amp;\cdots\\ 
-10   &amp; 30   &amp; -10  &amp; 1    &amp; 0    &amp;0     &amp;\cdots\\ 
-50   &amp; -10  &amp; 30   &amp; -10  &amp; 1    &amp;0     &amp;\cdots\\ 
-42   &amp; -50  &amp; -10  &amp; 30   &amp; -10  &amp;1     &amp;\cdots\\ 
\vdots&amp;\vdots&amp;\vdots&amp;\vdots&amp;\vdots&amp;\vdots&amp;\ddots 
\end{array}
\right] .
$$
By formula 
\eqref{eq:row}, 
the matrix for the row sums 
$\misSum A$ 
is
\begin{center}
$
\misSum A=
\left[
\begin{array}{rrrrrrrrrrrrrrrr}
1     &amp; 1    &amp;1     &amp;1     &amp;1     &amp;1     &amp;1     &amp;1     &amp;1     &amp;\cdots\\ 
-1    &amp; -3   &amp;-5    &amp;-7    &amp;-9    &amp; -11  &amp;-13   &amp; -15  &amp;-17   &amp;\cdots\\ 
-3    &amp; -3   &amp; 1    &amp; 9    &amp; 21   &amp; 37   &amp;57    &amp;81    &amp;109   &amp;\cdots\\ 
-5    &amp; 1    &amp; 11   &amp;  17  &amp; 11   &amp;-15   &amp; -69  &amp;-159  &amp;-293  &amp;\cdots\\
-7    &amp; 9    &amp; 17   &amp; 1    &amp; -39  &amp;-87   &amp;-111  &amp;-63   &amp;121   &amp;\cdots\\ 
-9    &amp; 21   &amp; 11   &amp; -39  &amp;-81   &amp; -51  &amp;99    &amp;369   &amp;679   &amp;\cdots\\ 
-11   &amp; 37   &amp; -15  &amp; -87  &amp;-51   &amp; 141  &amp;393   &amp;465   &amp;37    &amp;\cdots\\
-13   &amp; 57   &amp; -69  &amp;-111  &amp;99    &amp; 393  &amp;363   &amp;-351  &amp;-1709 &amp;\cdots\\
-15   &amp; 81   &amp; -159 &amp;-63   &amp;369   &amp; 465  &amp;-351  &amp;-1791 &amp;-2447 &amp;\cdots\\
-17   &amp; 109  &amp; -293 &amp;121   &amp;679   &amp; 37   &amp;-1709 &amp;-2447 &amp; 479  &amp;\cdots\\
\vdots&amp;\vdots&amp;\vdots&amp;\vdots&amp;\vdots&amp;\vdots&amp;\vdots&amp;\vdots&amp;\vdots&amp;\ddots
\end{array}
\right] .
$
\end{center}
Note that the removal of the first line of 
$\misSum A$ 
yields a symmetric matrix. 
Therefore, it follows from  formula 
\eqref{eq:bacher} 
and the formula for the generating function of the row sum sequence
$d(t)/(1-h(t))$ 
\cite{bijections,HeSha:sum} 
that 
$\rowsum n A=(1-3t)^n/(1-t)^{n+1}$ 
is the generating function for both the 
$n$th 
column and the 
$(n+1)$th 
row of 
$\misSum A$, 
where 
$n\geq1$. 
The first column 
(and the second row) 
of 
$\misSum A$ 
is the odd negative integers sequence 
\seqnum{A165747}, 
i.e., 
$(\misSum A)_{k,1}=1-2k$, 
$k\geq 1$. 
The second column 
(and the third row for $n\geq 3$)
is the pinwheel numbers sequence 
\seqnum{A059993}, 
i.e., 
$(\misSum A)_{k,2}=2(k-4)^2+6(k-4)+1$, 
$k\geq 4$.
\end{example}
\subsection{The array sequences $(\phi_{\beta,l}(\DD))_{l\geq1}$}
\label{sec:bacher}
We define infinite square arrays analogous to those of Section 
\ref{sec:matrices}. 
\begin{definition}
\label{def:infiniteB} 
Fix a number 
$\beta\in\CzeroP$.
For any Riordan array  
$\DD=(\dd(t),h(t))$ 
and any integer 
$i\geq 1$, 
define the following infinite matrices:
\begin{enumerate}[(i)]
\item 
Matrix 
$\Bmis i {\beta,\DD}$, 
whose 
$l$th 
column is the 
$i$th 
column of 
$\phi_{\beta,l}(\DD)$:
\begin{equation*}
\label{eq:Bcol}
(\Bmis i  {\beta,\DD})_{k,l}=
(\phi_{\beta,l}(\DD))_{k,i}=
[\te^{k+i-1}]\dd(\te)^\beta\h'(\te)^l\h(\te)^{i} , 
\quad 
k,l\geq1 .
\end{equation*}
\item 
Matrix  
$\BmisD i {\beta,\DD}$, 
whose 
$l$th 
column is the 
$i$th 
(descending) diagonal slice of  
$\phi_{\beta,l}(\DD)$:
\begin{equation*}
\label{eq:Bdiag}
(\BmisD i   {\beta,\DD})_{k,l}=
(\phi_{\beta,l}(\DD))_{k+i,k}=
[\te^{k+i-1}]\dd(\te)^\beta\h'(\te)^l\h(\te)^k , 
\quad 
k,l\geq1 .
\end{equation*}
\item 
Let 
$(\rowsum l {\beta,\DD})_m$, 
$m\geq 0$, 
be the sequence whose 
$m$th 
term is the sum of the entries of the 
$m$th 
row of  
$\phi_{\beta,l}(\DD)$, 
for 
$l\geq1$. 
In this context, we define the matrix 
$\BmisSum  {\beta,\DD}$, 
whose 
$l$th 
column is the sequence of the row sums of 
$\phi_{\beta,l}(\DD)$:
\begin{equation*}
\label{eq:Brow}
(\BmisSum  {\beta,\DD})_{k,l}=
(\rowsum l {\beta,\DD})_k=
[t^{k}]\frac{h(t)d(t)^\beta h'(t)^l}{(1-h(t))} , 
\quad 
k,l\geq1 .
\end{equation*}
\end{enumerate}
\end{definition}
Moreover, given a Riordan array 
$\DD$, 
analogous matrix definitions hold for the array sequences
$(\phi_{\beta,-l}(\DD))_{l\geq1}$ 
and 
$(\phi_{\beta,\pm1/l}(\DD))_{l\geq1}$.
\begin{example}
\label{ex:bacher}
\normalfont
We now examine some of the square matrices of Definition 
\ref{def:infiniteB} 
taking as base array the Pascal array   
$P=(p(t),tp(t))$, 
with 
$p(t)=1/(1-t)$. 
First, we fix 
$\beta=1$
and consider the matrix 
$\Bmis i {1,P}$. 
We have 
$$
(\Bmis i {1,P})_{k,l}=[\te^{k-1}]\frac{1}{(1-t)^{2l+i+1}} , 
\quad 
k,l\geq1 .
$$
Hence, the 
$l$th 
column of 
$\Bmis i {1,P}$ 
is  generated by 
$1/(1-t)^{2l+i+1}$ 
for any 
$l\geq 1$. 
Therefore,  
$$
(\Bmis i {1,P})_{k,l}= {2l+i+k-1\choose 2l+i}
                     = {2l+i+k-1\choose k-1} .
$$  
For 
$i=1$ 
we have
\begin{center}
$
\Bmis 1 {1,P}=
\left[
\begin{array}{llllllll}
1     &amp;1     &amp; 1    &amp;1     &amp;1     &amp;1     &amp;\cdots\\ 
4     &amp;6     &amp; 8    &amp;10    &amp;12    &amp;14    &amp;\cdots\\ 
10    &amp;21    &amp; 36   &amp;55    &amp;78    &amp;105   &amp;\cdots\\ 
20    &amp;56    &amp; 120  &amp;220   &amp;364   &amp;560   &amp;\cdots\\
35    &amp;126   &amp; 330  &amp;715   &amp;1365  &amp;2380  &amp;\cdots\\ 
56    &amp;252   &amp; 792  &amp;2002  &amp;4368  &amp;8568  &amp;\cdots\\ 
84    &amp;462   &amp; 1716 &amp;5005  &amp;12376 &amp;27132 &amp;\cdots\\
120   &amp;792   &amp; 3432 &amp;11440 &amp;31824 &amp;77520 &amp;\cdots\\
\vdots&amp;\vdots&amp;\vdots&amp;\vdots&amp;\vdots&amp;\vdots&amp;\ddots
\end{array}
\right] .
$
\end{center}
The sequences for the first columns are 
\seqnum{A000292}, 
\seqnum{A000389}, 
\seqnum{A000580}, 
\seqnum{A000582}, 
\seqnum{A001288}, 
and 
\seqnum{A010966}. 
From the  second row on the sequences for the listed rows are 
\seqnum{A005843}, 
\seqnum{A014105}, 
\seqnum{A002492}, 
\seqnum{A053126}, 
\seqnum{A053127},  
\seqnum{A053128}, 
and 
\seqnum{A053129}.   

Also for 
$\beta=1$,  
we have
$$
(\BmisSum{1,P})_{k,l}=
[\te^{k-1}]\frac{1}{(1-2t)(1-t)^{2l+1}} , 
\quad 
k,l\geq1 .
$$
The 
$l$th 
column of 
$\BmisSum {1,P}$ 
is thus generated by 
$1/((1-2t)(1-t)^{2l+1})$ 
for any 
$l\geq 1$. 
Therefore,  
$$
(\BmisSum 
{1,P})_{k,l} = \sum_{i=0}^{k-1} 2^i{2l+k-1-i\choose k-1-i}
             = 2^{k+2l}-\sum_{i=0}^{2l}{k+2l\choose i} .
$$
The matrix yields
\begin{center}
$
\BmisSum {1,P}=
\left[
\begin{array}{llllllll}
1     &amp;1     &amp;1     &amp; 1    &amp; 1     &amp;1     &amp;\cdots\\ 
5     &amp;7     &amp;  9   &amp; 11   &amp; 13    &amp;15    &amp;\cdots\\ 
16    &amp;29    &amp; 46   &amp; 67   &amp; 92    &amp;121   &amp;\cdots\\ 
42    &amp;93    &amp; 176  &amp; 299  &amp; 470   &amp;697   &amp;\cdots\\
99    &amp;256   &amp; 562  &amp; 1093 &amp; 1941  &amp;3214  &amp;\cdots\\ 
219   &amp;638   &amp; 1586 &amp; 3473 &amp; 6885  &amp; 12616&amp;\cdots\\
466   &amp;1486  &amp; 4096 &amp; 9949 &amp; 21778 &amp; 43796&amp;\cdots\\ 
968   &amp;3302  &amp; 26333&amp; 63004&amp; 137980&amp;280600&amp;\cdots\\
\vdots&amp;\vdots&amp;\vdots&amp;\vdots&amp;\vdots &amp;\vdots&amp;\ddots
\end{array}
\right] .
$
\end{center}
The sequences for the first four columns are 
\seqnum{A002662}, 
\seqnum{A002664}, 
\seqnum{A035039}, 
and 
\seqnum{A035041}.
The second and third rows are  
\seqnum{A005408} 
and 
\seqnum{A130883}.
From the  fourth row on and from 
$n$ 
such that 
$a(n)=2^n-\sum_{i=0}^2{n+2\choose i}$, 
$n\geq 4$, 
the sequences for the listed rows yield the alternating elements of the 
following sequences 
$a(n)$:
\seqnum{A000125}, 
\seqnum{A000127}, 
\seqnum{A006261},  
\seqnum{A008859}, 
and 
\seqnum{A008860}.
\end{example}
\section{Automorphisms and the $A$ and $Z$-sequences}
\label{sec:sequences}
Merlini, Sprugnoli, Rogers, and Verri 
\cite{MSRV}, 
and He and Sprugnoli 
\cite{HeSprug:seqChar}, 
showed that every 
Riordan array 
$\DD=(\dd(\te),\h(\te))$ 
can  be characterized by their 
$A$- 
and 
$Z$-sequences. 
We examine in this Section how the 
$A$- 
and 
$Z$-sequences of 
$\DD$ 
are transformed by the automorphisms 
$\phi_{\alpha,\beta}=\varphi_{\alpha,\beta,0}$. 
\begin{lemma}
[\cite{MSRV,HeSprug:seqChar,yang}]
\label{lem:AZ-sequences}
Let 
$\DD = (\dd(\te), \h(\te))$ 
be a Riordan array, and let 
$A(\te)$ 
and 
$Z(\te)$ 
be the generating functions for the corresponding 
$A$- 
and 
$Z$-sequences, 
respectively. Then, 
\begin{equation*}
\label{eq:d,h of A,Z}
\dd(t)=\frac{1}{1-tZ(\h(\te))} , 
\quad 
\h(\te)=\te A(\h(\te)) .
\end{equation*}
In terms of the inverse 
$\DD^{-1}=\left(1/\dd(\bar{h}(\te)),\bar{h}(\te)\right)$ 
we have
\begin{equation*}
\label{eq:A,Z invD}
A(\te)=\frac{\te}{\bar{\h}(\te)} 
\quad
\text{and}
\quad 
Z(\te)=\frac{A(\te)}{\te}\biggl(1-\frac{1}{\dd(\bar{\h}(\te))}\biggr) .
\end{equation*}
\end{lemma}
Moreover, let 
$(p_k(x))_{k\geq1}$ 
be the polynomial sequence associated with a Riordan array 
$\DD$, 
which is given by  
$p_k(x)=\sum_{n=1}^{k} \DD_{k,n} x^{n-1}$. 
By the multiplication rule for Riordan arrays 
\eqref{eq:multR}, 
the generating function of 
$(p_k(x))_{k\geq1}$ 
is
$p(\te,x)=\dd(\te)/(1-x\h(\te))$.
In this context, if 
$A_{\alpha,\beta}(\te)$ 
and 
$Z_{\alpha,\beta}(\te)$ 
denote the 
$A$- 
and 
$Z$-sequences 
of 
$\phi_{\alpha,\beta}(\DD)$, 
then a simple calculation yields
\begin{equation}
\label{eq:AZrs}
A_{\alpha,\beta}(\te)=A(\te)
\quad
\text{and}
\quad 
Z_{\alpha,\beta}(\te)=
\frac{A(\te)^{1-\alpha-\beta}}{t}
\biggl(
A(\te)^{\alpha+\beta} -\bigl(A(\te)-\te Z(\te)\bigr)^\beta
\biggr) .
\end{equation}
Likewise, the generating function 
$p_{\alpha,\beta}(\te,x)$
of the polynomial sequence of 
$\phi_{{\alpha},\beta}(\DD)$
is given by 
$p_{\alpha,\beta}(\te,x)=(\h(\te)/\te)^\alpha\dd(\te)^\beta/(1-x\h(\te))$. 
Now, Yang 
\cite{yang} 
introduced the function 
\begin{equation*}
\label{eq:B}
B(\te;x)=\frac{A(\te)-\te Z(\te)}{A(\te)-xt} ,
\end{equation*}
which satisfies 
\begin{equation*}
\label{eq:ftraB}
(\dd(\te),\h(\te))B(\te;x)=\frac{1}{1-xt} .
\end{equation*}
The function 
$B(\te;x)$ 
turns out to be the generating function of the polynomial sequence associated 
with the Riordan array 
$\DD^{-1}$.
Note that Yang's notation omits the variable  
$x$ 
in the definition of 
$B(\te;x)$, 
i.e.,
$B(\te;x)=B(\te)$ 
in \cite{yang}.
Now, if we define 
$
B_{\alpha,\beta}(\te;x)=
(A_{\alpha,\beta}(\te)-\te Z_{\alpha,\beta}(\te))/(A_{\alpha,\beta}(\te)-xt)
$,
this function reads in terms of 
$B(\te;x)$ 
as follows
\begin{eqnarray}
\nonumber
B_{\alpha,\beta}(\te;x) 
&amp; = &amp; 
A(\te)^{1-\alpha-\beta}(A(\te)-x\te)^{\beta-1}B(\te;x)^\beta\\
\label{eq:Brs}
&amp; = &amp; 
A(\te)^{-\alpha}B(\te;x)^\beta
\sum_{i=0}^{\beta-1}{\beta-1\choose i}A(\te)^{-i}(-x\te)^i .
\end{eqnarray}
In particular, for 
$\beta=1$
we have 
$B_{\alpha,1}(\te;x)=A(\te)^{-\alpha}B(\te;x)$.
\subsection{Infinite matrices associated with the $Z$ and 
$B(x)$-sequences} 
Let 
$\DD=(\dd(\te),\h(\te))$ 
be any Riordan array. For each matrix in the sequence 
$(\phi_{n}(\DD))_{n\geq 1}$, 
we denote by 
$Z_{n}^\DD$ 
and 
$B_{n}^\DD(x)$ 
the corresponding 
$Z$ 
and 
$B(x)$-sequences. 
Recall that the 
$A$-sequence 
is invariant under Bacher's automorphisms. By analogy with Section 
\ref{sec:infinite}, 
we define infinite matrices associated with the sequences 
$(Z_{n}^\DD(\te))_{n\geq 1}$ 
and 
$(B_{n}^\DD(\te;x))_{n\geq 1}$, 
where the sequences 
$Z_{n}^\DD(\te)$ 
and 
$B_{n}^\DD(\te;x)$ 
are defined in terms of the sequences 
$A,Z$, 
and 
$B(\te;x)$ 
of the matrix 
$\DD$ 
by formulas 
\eqref{eq:AZrs} 
and 
\eqref{eq:Brs}.
\begin{definition}
\label{def:infZB}
For any Riordan matrix 
$\DD$ 
and any real variable
$x$ 
define the following two infinite matrices:
\begin{equation}
\label{eq:ZBmatrices}
\ZZ \DD_{k,n}=[\te^{k-1}]Z_{n}^\DD
\quad
\text{and}
\quad
\BB \DD_{k,n}(x)= 
[\te^{k-1}]B_{n}^\DD(x) ,
\quad 
k,n\geq 1 .
\end{equation}
Hence, by definition the matrices 
$\ZZ \DD$ 
and 
$\BB \DD(x)$ 
are those whose 
$n$th 
column is generated by the sequences  
$Z$ 
and 
$B(x)$ 
of 
$\pp n (\DD)$.
\end{definition}
Again, analogous matrix definitions 
$\zz \DD$ 
and 
$\bb  \DD(x)$ 
hold
for the matrix sequence 
$(\pp {-n}(\DD))_{n\geq 1}$.
\begin{example}
\normalfont
We study the matrices 
$\ZZ P$ 
and 
$\BB P(x)$ defined by formula 
\eqref{eq:ZBmatrices}, 
for the Pascal array 
$P=(p(\te),\te p(\te))$, 
where 
$p(\te)=1/(1-\te)$ 
(see array \eqref{eq:pascal}). 
The Pascal triangle has the following well-known 
$A,Z$, 
and 
$B(x)$-sequences:
$$
A(\te)=1+\te ,
\quad 
Z(\te)=1 ,
\quad
\text{and}
\quad
B(\te;x)=\sum_{i=0}^\infty(-1+x)^i\te^i .
$$
The matrix 
$\ZZ P$ 
yields
\begin{center}
$
\ZZ P=
\left[
\begin{array}{rrrrrrrrrrrr}
2     &amp;3     &amp;4     &amp;5     &amp;6     &amp;7     &amp;\cdots\\ 
-1    &amp;-3    &amp;-6    &amp;-10   &amp;-15   &amp;-21   &amp;\cdots\\ 
1     &amp;4     &amp;10    &amp;20    &amp;35    &amp;56    &amp;\cdots\\ 
-1    &amp;-5    &amp;-15   &amp;-35   &amp;-70   &amp;-126  &amp;\cdots\\
1     &amp;6     &amp;21    &amp;56    &amp; 126  &amp;252   &amp;\cdots\\ 
-1    &amp;-7    &amp;-28   &amp;-84   &amp; 210  &amp;-462  &amp;\cdots\\ 
1     &amp;8     &amp;36    &amp;120   &amp; 330  &amp;792   &amp;\cdots\\
-1    &amp;-9    &amp;-45   &amp;-165  &amp;-495  &amp;-1287 &amp;\cdots\\
\vdots&amp;\vdots&amp;\vdots&amp;\vdots&amp;\vdots&amp;\vdots&amp;\ddots
\end{array}
\right] .
$
\end{center}
Then, by taking absolute values for the entries, the rows of  
$\ZZ P$ 
are the columns of the matrix 
$\pp 1 (P)$. 
More precisely,
$$
\ZZ P _{k,n}=(-1)^{k-1}{n+k-1\choose k} ,
\quad 
n\geq1 , k&gt;1 .
$$
 
Next, we consider the matrix $\BB P(x)$:
\begin{equation*}
\footnotesize
\BB P(x)=
\left[
\begin{array}{cccc}
1                   &amp;1                    &amp; 1                    &amp;\cdots\\ 
-2+x                &amp;-3+x                 &amp; -4+x                 &amp;\cdots\\  
3-3x+x^2            &amp;6-4x+x^2             &amp;10-5x+x^2             &amp;\cdots\\ 
-4+6x-4x^2+x^3      &amp;-10+10x-5x^2+x^3     &amp;-20+15x-6x^2+x^3      &amp;\cdots\\
5-10x+10x^2-5x^3+x^4&amp;15-20x+15x^2-6x^3+x^4&amp; 35-35x+21x^2-7x^3+x^4&amp;\cdots\\
\vdots              &amp;\vdots               &amp;\vdots                &amp;\ddots
\end{array}
\right] .
\end{equation*}
\normalsize
The entries 
$\BB P(x)$ 
are thus monic polynomials of degree 
$k-1$:
$$
\BB P_{k,n}(x)
=
\sum_{i=0}^{k-1}
(-1)^{k-1+i}{n+k-1\choose n+i}x^{i} ,
\quad
k,n\geq 1 .
$$
For 
$x=1$, 
the matrix 
$\BB P(1)$ 
is Pascal's array 
(read as a square matrix) 
with alternating signs:
\begin{center}
$
\BB P (1)=
\left[
\begin{array}{rrrrrrr}
1     &amp; 1    &amp;1     &amp;1     &amp;1     &amp;1     &amp;\cdots\\ 
-1    &amp; -2   &amp;-3    &amp;-4    &amp;-5    &amp;-6    &amp;\cdots\\ 
1     &amp; 3    &amp;6     &amp; 10   &amp;15    &amp;21    &amp;\cdots\\ 
-1    &amp; -4   &amp;-10   &amp;-20   &amp;-35   &amp;-56   &amp;\cdots\\
1     &amp; 5    &amp;15    &amp; 35   &amp;70    &amp;126   &amp;\cdots\\ 
-1    &amp; -6   &amp;-21   &amp;-56   &amp;-126  &amp;-462  &amp;\cdots\\ 
1     &amp; 7    &amp;28    &amp; 84   &amp;330   &amp;792   &amp;\cdots\\
-1    &amp; -8   &amp;-36   &amp;-120  &amp;-495  &amp;-1287 &amp;\cdots\\
\vdots&amp;\vdots&amp;\vdots&amp;\vdots&amp;\vdots&amp;\vdots&amp;\ddots
\end{array}
\right] .
$
\end{center}
\end{example}
\section{Automorphisms and Riordan group pseudo-involutions}
\label{sec:inv}
We study the action of Bacher's automorphisms on the 
pseudo-involutions of the Riordan group.

We recall that a Riordan array 
$\DD=(\dd(\te),\h(\te))$ 
is an involution if and only if 
\begin{eqnarray*}
\label{eq:order 2 d}
\dd(\te)    &amp; = &amp; {1\over{\dd(\h(\te))}} ,\\
\label{eq:order 2 h}
\h(\h(\te)) &amp; = &amp;\te .
\end{eqnarray*}
Note that for any involution 
$I$ 
and any Riordan array 
$\DD$, 
the identity 
$(\DD I)^2=\id{\Rio}$ 
is equivalent to 
$(I\DD)^2=\id{\Rio}$. 
Therefore, since 
$\DD$ 
has an inverse, we have 
$(\DD I)^2=\id{\Rio}$ 
if and only if 
$(\DD^nI)^2=\id{\Rio}$ 
for any integer 
$n$.
In this context, Shapiro 
\cite{sha:open} 
and later Cameron and Nkwanta 
\cite{cameron46} 
introduced the following notion of pseudo-involution.
An element 
$\DD$ 
of the Riordan group has 
{\em pseudo-order 2} 
if the matrix
$\DD M$ 
has order 2, where 
$M=(1,-\te)$
is an involution. Therefore, a Riordan array 
$\DD=(\dd(\te),\h(\te))$
with pseudo-order 2 is known as a
{\em pseudo-involution}. 
In this case, it readily follows that
\begin{eqnarray}
\label{eq:pseudo-order d}
\dd(\te)     &amp; = &amp;{1\over{\dd(-\h(\te))}} ,\\
\label{eq:pseudo-order h}
\h(-\h(\te)) &amp; = &amp;-\te .
\end{eqnarray}
In particular, if 
$\DD=(1,h(t))$
is an element of the Lagrange subgroup such that
\begin{equation*}
\label{eq:pseudo-order hII}
\h(\te) = \bar{\h}(\te)
\quad
\text{and}
\quad 
\h(-\te)=-\h(\te) ,
\end{equation*}
then 
$\DD$ 
has pseudo-order 2 
\cite{cameron46}.
We next give the following characterization of pseudo-involutions.
\begin{proposition}
\label{prop:pinv} 
Let 
$\alpha:\Rio\to \Rio$ 
be any Riordan group automorphism. If  
$\alpha(M)=M$, 
then an element 
$\DD\in\Rio$ 
has pseudo-order 2 if and only if 
$\alpha(\DD)$ 
has pseudo-order 2.
\end{proposition}
\begin{proof} 
Suppose that 
$\DD M$ 
has order 2. Then,
$(\alpha(\DD)M)^2=\alpha((\DD M)^2)=\alpha(\id{\Rio})=\id{\Rio}$. 
Suppose now that 
$\alpha(\DD)$ 
has pseudo-order 2. Then, 
$\id{\Rio}=(\alpha(\DD)M)^2=\alpha((\DD M)^2)$. 
Therefore, 
$(\DD M)^2=\id{\Rio}$, 
for 
$\alpha$ 
is injective.
\end{proof}
Note that the result above does not use the Riordan group 
structure.  Therefore, the proposition remains valid for a generalization of 
the concept of pseudo-involution to any group.
Furthermore, since 
$\varphi_{\alpha,\beta,\gamma}(M)=((-1)^{\alpha+\gamma},-\te)$,
not all the automorphisms 
$\varphi_{\alpha,\beta,\gamma}$  
leave the Riordan array 
$M$ 
invariant. Nevertheless, they all preserve pseudo-involutions as we show
next.
\begin{proposition}
\label{prop:inv prs} 
For any  
$\alpha,\gamma\in\C$ 
and 
$\beta\in\C^*$, 
a Riordan matrix 
$\DD\in\Rio$ 
has pseudo-order 2 if and only if 
$\varphi_{\alpha,\beta,\gamma}(\DD)$
has pseudo-order 2. 
\end{proposition}
\begin{proof}
Suppose that 
$\DD=(\dd(\te),\h(\te))$
has pseudo-order 2.  By equations 
\eqref{eq:pseudo-order d} 
and 
\eqref{eq:pseudo-order h} 
we have
\begin{eqnarray*}
(\varphi_{\alpha,\beta,\gamma}(\DD M)^2 
&amp; = &amp; 
\left
(\biggl(\frac{\h(\te)}{\te}\biggr)^\alpha\dd(\te)^\beta h'(t)^\gamma,-\h(\te)
\right)^2\\
&amp; = &amp; 
\left(
\biggl(\frac{\h(\te)}{\te}\biggr)^\alpha
\dd(\te)^\beta h'(t)^\gamma
\biggl(\frac{\h(-\h(\te))}{-\h(\te)}\biggr)^\alpha
\dd(-\h(\te))^\beta h'(-h(t))^\gamma,\te
\right)\\
&amp; = &amp; (1,\te) .
\end{eqnarray*}
Now, suppose that 
$\prs(\DD)$ 
has pseudo-order 2. From equation
\eqref{eq:pseudo-order d} 
we have 
\begin{eqnarray*}
\biggl(\frac{\h(\te)}{\te}\biggr)^\alpha
\dd(\te)^\beta h'(t)^\gamma
&amp;=&amp;
\biggl(\frac{-\h(\te)}{\h(-\h(\te))}\biggr)^\alpha
\frac{1}{\dd(-\h(\te))^\beta h'(-h(t))^\gamma} .
\end{eqnarray*}
Using 
$\h(-\h(\te))=-\te$  
from equation
\eqref{eq:pseudo-order h} 
gives 
$\dd(\te)=1/\dd(-\h(\te))$.
\end{proof}
This proposition extends Cheon, Kim, Jin, and Shapiro's remark that says that 
if 
$(\h(\te)/\te,\h(\te))$ 
is a pseudo-involution, then 
$((\h(\te)/\te)^n, \h(\te))$ 
is also a pseudo-involution for all integers 
$n$  
\cite{cheon}.
\begin{example}
\label{ex:rna}
\normalfont
Consider the generalized RNA triangles introduced by Cameron and Nkwanta 
\cite{cameron46} 
for   
$n\geq1$:
\begin{equation*}
\label{eq:Dn}
\Dpi_{\pm n}=
\left( 
g
\left(\te\right) 
\left(\frac{1-\te}{1-\te g\left(\te\right)}\right)^{\pm n},
\te g
\left(\te\right) 
\right) ,
\end{equation*}
where 
\begin{equation*}
g\left( \te\right)=
\frac{1-\te+\te^{2}-\sqrt{1-2\te-\te^{2}-2\te^{3}+\te^{4}}}{2\te^{2}}. 
\label{rna}
\end{equation*}
We recall that 
$\Dpi_{\pm n}=A^{\mp n}\Dpi_{0}A^{\pm n}$, 
where 
$\Dpi_{0}=\left(g\left(\te\right),\te g\left(\te\right) \right)$ 
is the Nkwanta RNA triangle, and 
$A^{\pm n}=\left(\frac{1}{\left( 1-\te\right) ^{\pm n}},\te\right)$. 
We now give an alternative definition of the matrices 
$\Dpi_{\pm n}$
in terms of 
the 
automorphisms 
$\pp{\pm n}$ 
for 
$n\geq 1$. 
Let 
\begin{equation}
\label{eq:D}
\Dpi=
\left(  
\frac{1-\te}{1-\te g(\te) },\te g(\te)
\right) ,
\end{equation}
\begin{center}
$
\Dpi=
\left[
\begin{array}{lllllll}
1     &amp; 0    &amp; 0    &amp; 0    &amp; 0    &amp; 0    &amp;\cdots\\ 
0     &amp; 1    &amp; 0    &amp; 0    &amp; 0    &amp; 0    &amp;\cdots\\ 
1     &amp; 1    &amp; 1    &amp; 0    &amp; 0    &amp; 0    &amp;\cdots\\ 
2     &amp; 2    &amp; 2    &amp; 1    &amp; 0    &amp; 0    &amp;\cdots\\ 
5     &amp; 5    &amp; 4    &amp; 3    &amp; 1    &amp; 0    &amp;\cdots\\ 
12    &amp; 12   &amp; 10   &amp; 7    &amp; 4    &amp; 1    &amp;\cdots\\ 
\vdots&amp;\vdots&amp;\vdots&amp;\vdots&amp;\vdots&amp;\vdots&amp;\ddots
\end{array}
\right] .
$
\end{center}
The diagonal slices of the array 
$\Dpi$ 
are of combinatorial interest. The sequence numbers for the first three
nontrivial diagonals are
\seqnum{A000124},
\seqnum{A177787}, 
and	
\seqnum{A116722}.
Moreover, the matrix 
$\Dpi$ 
given by formula 
\eqref{eq:D} 
is the base matrix for the sequences 
$(\Dpi_{\pm n})_{n\geq1}$. 
That is, for 
$n\geq1$ 
we have
\begin{equation*}
\label{eq:Dn2}
\Dpi_{\pm n}=\pp{\pm n}(\Dpi) .
\end{equation*}
Also,  the matrices 
$\phi_{\alpha,\beta}(\Dpi_{\pm n})$   
can be expressed in terms  of the base matrix 
$\Dpi$ 
as follows
\begin{equation*}
\label{eq:aut Dn}
\phi_{\alpha,\beta}(\Dpi_{\pm n})=
(\phi_{\alpha,\beta}\circ\pp{\pm n})(\Dpi)=
\phi_{ \alpha+\beta,({\pm n})\beta } (\Dpi) .
\end{equation*}
Note that the matrix 
$\Dpi$ 
is not a pseudo-involution. Indeed,
\begin{equation*}
(\Dpi M)^2=\biggl(\frac{1-t}{1-tg(t)},-tg(t)\biggr)^2
=\biggl(\frac{1-t}{1-tg(t)}\frac{1+tg(t)}{1+t},t\biggr)\neq(1,t) ,
\end{equation*}
since 
$g(t)$ 
satisfies the functional equation 
$g(t)g(-tg(t))=1$.
Therefore, it follows directly from Proposition 
\ref{prop:inv prs} 
that the matrices 
$\Dpi_{\pm n}$
are not pseudo-involutions 
either, 
which contradicts Theorem 3 of 
\cite{cameron46}. 
Observe also that although the aforementioned definition of the generalized RNA 
matrices, namely,
$$
\Dpi_{\pm n}=
A^{\mp n}\Dpi_{0}A^{\pm n}=\big((1-t)^{\pm n},t\big)\big(g(t),tg(t)\big)
\left(\frac{1}{\left( 1-\te\right) ^{n}},\te\right)
$$
given in \cite{cameron46} does not yield pseudo-involutions, we 
can easily define pseudo-involutions from Nkwanta's RNA matrix by setting
$$
\Dpi^{\pm}_{\pm n}=
\Big((1\pm t)^{\pm n},t\Big)
\Dpi_{0}\left(\frac{1}{\left( 1\mp \te\right)^{\pm n}},\te\right)
=
\Big(g(t)\frac{1\pm t}{1\mp tg(t)},tg(t)\Big) ,
\quad 
n\geq1 .
$$
It is worth examining some of the matrices 
$\mis i \Dpi$, 
$\misD i \Dpi$, 
and 
$\misSum \Dpi$ 
(recall formulas 
\eqref{eq:col}, 
\eqref{eq:diag}, and 
\eqref{eq:row} 
of Section 
\ref{sec:matrices}). 
First, we study the matrix 
$\mis 1 \Dpi$ 
whose columns are given by the first column of each of the matrices 
$\Dpi_n$ 
for 
$n\geq1$. 
We have
\begin{center}
$
\mis 1 \Dpi=
\left[
\begin{array}{llllllll}
1     &amp; 1    &amp; 1    &amp; 1    &amp; 1    &amp; 1    &amp;\cdots\\ 
1     &amp; 1    &amp; 1    &amp; 1    &amp; 1    &amp; 1    &amp;\cdots\\ 
2     &amp; 3    &amp; 4    &amp; 5    &amp; 6    &amp; 7    &amp;\cdots\\ 
5     &amp; 8    &amp; 11   &amp; 14   &amp; 17   &amp; 20   &amp;\cdots\\
12    &amp; 21   &amp; 31   &amp; 42   &amp; 54   &amp; 67   &amp;\cdots\\ 
29    &amp; 55   &amp; 86   &amp; 122  &amp; 163  &amp; 209  &amp;\cdots\\ 
71    &amp; 144  &amp; 237  &amp; 351  &amp; 487  &amp; 646  &amp;\cdots\\
175   &amp; 377  &amp; 650  &amp;1001  &amp;1437  &amp; 1965 &amp;\cdots\\
434   &amp; 987  &amp; 1775 &amp;2833  &amp;4197  &amp; 5904 &amp;\cdots\\
\vdots&amp;\vdots&amp;\vdots&amp;\vdots&amp;\vdots&amp;\vdots&amp;\ddots
\end{array}
\right] .
$
\end{center}
The second column is the sequence 
\seqnum{A088305}. 
Moreover, the sequences for the first nontrivial rows are
\begin{itemize}
\item 
$(\mis 1 \Dpi) _{4,n}=2+3n$, 
$n\geq 1$; 
\item 
$(\mis 1 \Dpi)_{5,1}=12$ 
and 
$(\mis 1 \Dpi)_{5,n}=(\mis 1 \Dpi)_{5,n-1}+7+n$, 
$n&gt;1$;
\item  
$(\mis 1 \Dpi)_{6,1}=29$ 
and 
$(\mis 1 \Dpi)_{6,n}=(\mis 1 \Dpi)_{6,n-1}+16+5n$, 
$n&gt;1$;
\item 
$(\mis 1 \Dpi)_{7,1}=71$, 
$(\mis 1 \Dpi)_{7,2}=144$,  
and 
$(\mis 1\Dpi)_{7,n}=2(\mis 1 \Dpi)_{7,n-1}-(\mis 1 \Dpi)_{7,n-2}+17+n$,  
$n&gt;2$;
\item 
$(\mis 1 \Dpi)_{8,1}=175$, 
$(\mis 1 \Dpi)_{8,2}=377$,  
and 
$(\mis 1 \Dpi)_{8,n}=2(\mis 1 \Dpi)_{8,n-1}-(\mis 1 \Dpi)_{8,n-2}+50+7n$, 
$n&gt;2$;
\item 
$(\mis 1 \Dpi)_{9,1}=434$, 
$(\mis 1 \Dpi)_{9,2}=987$,  
and 
$(\mis 1 \Dpi)_{9,n}=2(\mis 1 \Dpi)_{9,n-1}-(\mis1 \Dpi)_{9,n-2}
+136+(n^2+63n)/2$,
$n&gt;2$.
\end{itemize}
We now analyse the first nontrivial diagonal slices of the 
matrices 
$\Dpi_n$. 
The matrix of the third diagonal slices reads as follows
\begin{center}
$
\misD 3 \Dpi=
\left[
\begin{array}{llllllll}
2     &amp; 3    &amp; 4    &amp; 5    &amp; 6    &amp; 7    &amp;\cdots\\ 
4     &amp; 5    &amp; 6    &amp; 7    &amp; 8    &amp; 9    &amp;\cdots\\ 
7     &amp; 8    &amp; 9    &amp; 10   &amp; 11   &amp; 12   &amp;\cdots\\ 
11    &amp; 12   &amp; 13   &amp; 14   &amp; 15   &amp; 16   &amp;\cdots\\
16    &amp; 17   &amp; 18   &amp; 19   &amp; 20   &amp; 21   &amp;\cdots\\ 
22    &amp; 23   &amp; 24   &amp; 25   &amp; 26   &amp; 27   &amp;\cdots\\ 
29    &amp; 30   &amp; 31   &amp; 32   &amp; 33   &amp; 34   &amp;\cdots\\
37    &amp; 38   &amp; 39   &amp;40    &amp;41    &amp; 42   &amp;\cdots\\
\vdots&amp;\vdots&amp;\vdots&amp;\vdots&amp;\vdots&amp;\vdots&amp;\ddots
\end{array}
\right] .
$
\end{center}
The entries are given by
\begin{equation*}
\label{eq: col D 3}
(\misD 3 \Dpi)_{k,n}= n+1+\frac{(k+2)^2-3(k+2)}{2},
\quad 
k,n\geq 1 .
\end{equation*}
Also, the following recursive definition holds
\begin{eqnarray*}
\label{eq: rec col D 3}
(\misD 3 \Dpi)_{1,n}&amp;=&amp; n+1 ,\\
(\misD 3 \Dpi)_{k,n}&amp;=&amp;(\misD 3 \Dpi)_{k-1,n}+k-1 ,
\quad 
k&gt;1 .
\end{eqnarray*}
The first five sequence numbers are 
\seqnum{A000124}, 
\seqnum{A152948}, 
\seqnum{A152950}, 
\seqnum{A145018}, and
\seqnum{A167499}.

Furthermore, as for the fourth diagonal slices of the matrices 
$\Dpi_n$ 
we have
\begin{center}
$
\misD 4 \Dpi=
\left[
\begin{array}{llllllll}
5     &amp; 8    &amp; 11   &amp;  14  &amp; 17   &amp; 20   &amp;\cdots\\ 
10    &amp; 14   &amp;  18  &amp; 22   &amp;  26  &amp; 30   &amp;\cdots\\ 
18    &amp; 23   &amp; 28   &amp; 33   &amp; 38   &amp; 43   &amp;\cdots\\ 
30    &amp; 36   &amp; 42   &amp;  48  &amp; 54   &amp; 60   &amp;\cdots\\
47    &amp; 54   &amp; 61   &amp; 68   &amp; 75   &amp; 82   &amp;\cdots\\ 
70    &amp; 78   &amp; 86   &amp; 94   &amp; 102  &amp; 110  &amp;\cdots\\ 
100   &amp; 109  &amp; 118  &amp; 127  &amp; 136  &amp; 145  &amp;\cdots\\
138   &amp; 148  &amp; 158  &amp;168   &amp;178   &amp; 188  &amp;\cdots\\
\vdots&amp;\vdots&amp;\vdots&amp;\vdots&amp;\vdots&amp;\vdots&amp;\ddots
\end{array}
\right] .
$
\end{center}
The first column is of course the fourth diagonal slice  of 
$\pp1(\Dpi)$ 
(and of the base matrix $\Dpi$),  
sequence 
\seqnum{A177787}. 
The second column is the sequence 
\seqnum{A201347}. 
The entries of 
$\misD 4 \Dpi$ 
are given by 
$$
(\misD 4 \Dpi)_{k,n}=\frac{(k+1)(11+(k+1)^2)}{2}+(2+k)(n-1) , 
\quad 
k,n\geq 1 .
$$
Equivalently, the sequence for the 
$k$th row, 
$k\geq 1$, 
yields
\begin{eqnarray*}
\label{eq: row D 4}
(\misD 4 \Dpi)_{k,n}&amp;=&amp;(\misD 4 \Dpi)_{k,n-1}+k+2,\quad n&gt;1 .
\end{eqnarray*}
The recursion formula for the $n$th column, $n\geq 1$, is
\begin{eqnarray*}
\label{eq: col D 4}
(\misD 4 \Dpi)_{1,1}&amp;=&amp; 5 ;\\
(\misD 4 \Dpi)_{1,n}&amp;=&amp;(\misD 4 \Dpi)_{1,n-1}+3, 
\quad 
n &gt;1 ,\\
(\misD 4 \Dpi)_{k,n}&amp;=&amp;(\misD 4 \Dpi)_{k-1,n}+n+k+6, 
\quad 
k&gt;1 .
\end{eqnarray*}

We next study the matrix 
$\misSum \Dpi$ 
for the row sum sequences of the matrices 
$\Dpi_n$:
\begin{center}
$
\misSum \Dpi=
\left[
\begin{array}{llllllll}
1     &amp; 1    &amp; 1    &amp; 1    &amp; 1    &amp; 1    &amp;\cdots\\ 
2     &amp; 2    &amp; 2    &amp; 2    &amp; 2    &amp; 2    &amp;\cdots\\ 
5     &amp; 6    &amp; 7    &amp; 8    &amp; 9    &amp; 10   &amp;\cdots\\ 
13    &amp; 17   &amp; 21   &amp; 25   &amp; 29   &amp; 33   &amp;\cdots\\
34    &amp; 48   &amp; 63   &amp; 79   &amp; 96   &amp; 114  &amp;\cdots\\ 
89    &amp; 134  &amp; 185  &amp; 242  &amp; 305  &amp; 374  &amp;\cdots\\ 
233   &amp; 371  &amp; 536  &amp; 729  &amp; 951  &amp; 1203 &amp;\cdots\\
610   &amp; 1021 &amp; 1537 &amp;2166  &amp;2916  &amp; 3795 &amp;\cdots\\
\vdots&amp;\vdots&amp;\vdots&amp;\vdots&amp;\vdots&amp;\vdots&amp;\ddots\\
\end{array}
\right] .
$
\end{center}
The  first column is the sequence 
\seqnum{A001519}:
\begin{eqnarray*}
\label{eq: 1col sum}
(\misSum  \Dpi)_{1,1} &amp; = &amp; 1 ,\\
(\misSum \Dpi)_{2,1}  &amp; = &amp;1 ,\\
(\misSum  \Dpi)_{k,n} &amp; = &amp;3(\misSum \Dpi)_{k-1,1}-(\misSum \Dpi)_{k-2,1}, 
\quad 
k&gt;1 .
\end{eqnarray*}
Also, the sequences for the first nontrivial rows are
\begin{itemize}
\item 
$(\misSum  \Dpi) _{4,n}=9+4n$, 
$n\geq 1$; 
\item 
$(\misSum  \Dpi)_{5,1}=34$ 
and 
$(\misSum \Dpi)_{5,n}=(\misSum \Dpi)_{5,n-1}+12+n$, 
$n&gt;1$;
\item 
$(\misSum \Dpi)_{6,1}=89$ 
and 
$(\misSum \Dpi)_{6,n}=(\misSum \Dpi)_{6,n-1}+33+6n$, 
$n&gt;1$;
\item
$(\misSum \Dpi)_{7,1}=233$, 
$(\misSum \Dpi)_{8,2}=371$, 
and \\ 
$(\misSum \Dpi)_{7,n}=2(\misSum \Dpi)_{7,n-1}-(\misSum \Dpi)_{7,n-2}+24+n$,
$n&gt;2$;
\item
$(\misSum \Dpi)_{8,1}=610$, $(\misSum \Dpi)_{8,2}=1021$, 
and 
\\ $(\misSum \Dpi)_{8,n}=2(\misSum \Dpi)_{8,n-1}-(\misSum \Dpi)_{8,n-2}+81+8n$, 
$n&gt;2$.
\end{itemize}
\end{example}
\begin{example}
\normalfont
We study some of the matrices for the diagonal slices of Section 
\ref{sec:bacher} taking as base 
the matrix 
$\Dpi_0=(g(t),tg(t))$.
For fixed 
$\beta\in\CzeroP$, 
the sequences 
$(D^\beta_{\pm l})_{l\geq1}=(\phi_{\beta,\pm l}(\Dpi_0))_{l\geq 1}$ 
are sequences of pseudo-involutions by Proposition 
\ref{prop:inv prs}. 
For the matrix 
$\Dpi_0$,
the second formula of Definition 
\ref{def:infiniteB} 
yields
$$
(\BmisD i   
{\beta,\Dpi_0})_{k,l}=[\te^{i-1}]\big((tg)'(\te)\big)^lg(\te)^{k+\beta}, 
\quad 
k,l\geq1 .
$$
Hence, restricting 
$\beta$ 
to integers 
$n\neq0$, 
the matrices 
$\BmisD i  {n+1,\Dpi_0}$ 
are obtained from 
$\BmisD i {n,\Dpi_0}$ 
by removing the first row. However, the matrix for 
$n=1$ 
is obtained from that for 
$n=-1$ 
by removing the first two rows. This is true for all the elements of the Bell 
subgroup. Thus, we now consider only 
$\beta=1$. 
The value 
$i=2$ 
defines the first nontrivial square matrix:
\begin{center}
$
\BmisD 2 {1,\Dpi_0}=
\left[
\begin{array}{llllllll}
4     &amp; 6    &amp; 8    &amp; 10   &amp; 12   &amp; 14   &amp;\cdots\\ 
5     &amp; 7    &amp; 9    &amp; 11   &amp; 13   &amp; 15   &amp;\cdots\\ 
6     &amp; 8    &amp; 10   &amp; 12   &amp; 14   &amp; 16   &amp;\cdots\\ 
7     &amp; 9    &amp; 11   &amp; 13   &amp; 15   &amp; 17   &amp;\cdots\\
8     &amp; 10   &amp; 12   &amp; 14   &amp; 16   &amp; 18   &amp;\cdots\\ 
9     &amp; 11   &amp; 13   &amp; 15   &amp; 17   &amp; 19   &amp;\cdots\\ 
10    &amp; 12   &amp; 14   &amp; 16   &amp; 18   &amp; 20   &amp;\cdots\\
11    &amp; 13   &amp; 15   &amp; 17   &amp; 19   &amp; 21   &amp;\cdots\\ 
\vdots&amp;\vdots&amp;\vdots&amp;\vdots&amp;\vdots&amp;\vdots&amp;\ddots
\end{array}
\right] .
$
\end{center}
Hence, we have
$$
(\BmisD 2 {1,\Dpi_0})_{k,l}=k+2l+1 ,
\quad 
k,l\geq 1 .
$$
Now, for 
$i=3$ 
the matrix for the diagonal slices of 
$\Dpi^1_l$ 
yields
\begin{center}
$
\BmisD 3 {1,\Dpi_0}=
\left[
\begin{array}{llllllll}
10    &amp; 21   &amp; 36   &amp; 55   &amp; 78   &amp; 105  &amp;\cdots\\ 
15    &amp; 28   &amp; 45   &amp; 66   &amp; 91   &amp; 120  &amp;\cdots\\ 
21    &amp; 36   &amp; 55   &amp; 78   &amp; 105  &amp; 136  &amp;\cdots\\ 
28    &amp; 45   &amp; 66   &amp; 91   &amp; 120  &amp; 153  &amp;\cdots\\
36    &amp; 55   &amp; 78   &amp; 105  &amp; 136  &amp; 171  &amp;\cdots\\ 
45    &amp; 66   &amp; 91   &amp; 120  &amp; 153  &amp; 190  &amp;\cdots\\ 
55    &amp; 78   &amp; 105  &amp; 136  &amp; 171  &amp; 210  &amp;\cdots\\
66    &amp; 91   &amp; 120  &amp; 153  &amp; 190  &amp; 231  &amp;\cdots\\ 
\vdots&amp;\vdots&amp;\vdots&amp;\vdots&amp;\vdots&amp;\vdots&amp;\ddots
\end{array}
\right] .
$
\end{center} 
The sequence for the odd rows is 
\seqnum{A014105}, 
while that for the even rows is 
\seqnum{A000384}. 
As for the columns, the sequence is 
\seqnum{A000217}. 
We have
$$
(\BmisD 3 
{1,\Dpi_0})_{k,l}={k+2l+2\choose 2} ,
\quad 
k,l\geq 1 .
$$
The next matrix for the diagonal slices of  
$\Dpi^1_l$ 
is
\begin{center}
$
\BmisD 4 {1,\Dpi_0}=
\left[
\begin{array}{llllllll}
26    &amp; 66   &amp; 134  &amp; 238  &amp; 386  &amp; 586  &amp;\cdots\\ 
42    &amp; 95   &amp; 180  &amp; 305  &amp; 478  &amp; 707  &amp;\cdots\\ 
64    &amp; 132  &amp; 236  &amp; 384  &amp; 584  &amp; 844  &amp;\cdots\\ 
93    &amp; 178  &amp; 303  &amp; 476  &amp; 705  &amp; 998  &amp;\cdots\\
130   &amp; 234  &amp; 382  &amp; 582  &amp; 842  &amp; 1170 &amp;\cdots\\ 
176   &amp; 301  &amp; 474  &amp; 703  &amp; 996  &amp; 1361 &amp;\cdots\\ 
232   &amp; 380  &amp; 580  &amp; 840  &amp; 1168 &amp; 1572 &amp;\cdots\\
299   &amp; 472  &amp; 701  &amp; 994  &amp; 1359 &amp; 1804 &amp;\cdots\\ 
\vdots&amp;\vdots&amp;\vdots&amp;\vdots&amp;\vdots&amp;\vdots&amp;\ddots
\end{array}
\right] .
$
\end{center}
The sequences for the first two columns are 
\seqnum{A000125} 
and 
\seqnum{A060163}.
The equation for the entries is 
\begin{equation*}
(\BmisD 4 {1,\Dpi_0})_{k,l} =  
\frac{\big(k+2(l+1)\big)^3+5\big(k+2(l+1)\big)+6(2l-1)}{6} ,
\quad 
k,l\geq 1 .
\end{equation*}
\end{example}
\section{Acknowledgments}
We would like to thank the anonymous referee for his valuable 
suggestions.
This research was made within the activities of the Group for Linear, Algebraic 
and Combinatorial Structures of the Center for Functional Analysis, Linear 
Structures and Applications (University of Lisbon, Portugal).
The first and the second authors were supported by the research fellowships 
SFRH/BPD/100567/2014 and SFRH/BPD/111317/2015, respectively, provided by the 
Portuguese Foundation for Science and Technology (FCT). 
\bibliographystyle{jis}
\begin{thebibliography}{1}
\bibitem{bacher}
R. Bacher, Sur le groupe d'interpolation, preprint, 2006. Available at 
\url{https://arxiv.org/abs/math/0609736}.
\bibitem{cameron46} 
N. T. Cameron and A. Nkwanta,  On some (pseudo) 
involutions in the Riordan group,  \emph{J. Integer Sequences}
\textbf{8} (2005),
\href{https://cs.uwaterloo.ca/journals/JIS/VOL8/Cameron/cameron46.pdf} {Article 
05.3.7}.
\bibitem{CKS:inv} 
G.-S. Cheon, H. Kim, and L. W. Shapiro,  Riordan 
group involutions, {\em Linear Algebra Appl.} \textbf{428} (2008)   941--952. 
\bibitem{cheon}
G.-S. Cheon, S.-T. Jin, H. Kim, and L. W. Shapiro,
Riordan group involutions and the $\Delta$-sequence,  {\em Discrete Appl. 
Math.} \textbf{157} (2009) 1696--1701.
\bibitem{hitt}
G.-S. Cheon, H. Kim, and L. W. Shapiro,   The hitting time 
subgroup, \L ukasiewicz paths and
Faber polynomials, {\em European J. Combin.} \textbf{32} (2011) 82--91.
\bibitem{CheonJin}
G.-S. Cheon and S.-T. Jin, Structural properties of Riordan 
matrices and extending the matrices, {\em Linear Algebra Appl.} \textbf{435}  
(2011) 
2019--2032.
\bibitem{HHS:RioGvsSheffG}
T.-X. He, L.~C. Hsu, and P.~J.-S. Shiue,
 The {S}heffer group and the {R}iordan group,
{\em Discrete Appl. Math.} \textbf{155} (2007) 1895--1909.
\bibitem{HeSha:sum} 
T.-X. He and L. W. Shapiro,
Row sums and alternating sums of Riordan arrays,
{\em Linear Algebra Appl.} \textbf{507} (2016) 77--95.
\bibitem{HeSprug:seqChar}
T.-X. He and R. Sprugnoli, 
Sequence characterization of Riordan arrays,
{\em Discrete Math.} \textbf{309} (2009) 3962--3974. 
\bibitem{jean-louis}
C. Jean-Louis and A. Nkwanta,
Some algebraic structure of the Riordan group,  {\em Linear Algebra Appl.}
\textbf{438} (2013) 2018--2035.
\bibitem{LM}
A. Luz\'on and M. A. Mor\'on,  Recurrence relations for 
polynomial sequences via Riordan matrices, {\em Linear Algebra 
Appl.}  \textbf{433} (2010) 1422--1446.
\bibitem{LMM}
A. Luz\'on, M. A. Mor\'on, and L. F. Prieto-Mart\'inez, A formula 
to construct all involutions in Riordan matrix groups, {\em Linear Algebra 
Appl.} \textbf{533} (2017) 397--417.
\bibitem{idInduced}
A. Luz\'on, D. Merlini, M. A. Mor\'on, and R. Sprugnoli, 
Identities
induced by Riordan arrays, {\em Linear Algebra Appl.}
\textbf{436} (2012)  631--647.
\bibitem{complementary}
A. Luz\'on, D. Merlini,
M. A. Mor\'on, and R. Sprugnoli, Complementary Riordan 
arrays, {\em Discrete 
Appl. Math.} \textbf{172} (2014) 75--87.
\bibitem{MSRV}
D. Merlini, D. G. Rogers, R. Sprugnoli, and M. C. 
Verri,  On
some alternative characterizations of Riordan arrays, {\em Canad.
J.  Math.} \textbf{49} (1997) 301--320.
\bibitem{merlini-verri}
D.~Merlini and M.~C. Verri,
 Generating trees and proper {R}iordan arrays,
{\em Discrete Math.} \textbf{218} (2000) 167--183
\bibitem{pw}
 P. Peart and W. Woan, A divisibility property for a subgroup 
of Riordan matrices, {\em Discrete Appl. Math.} \textbf{3} (2000) 121--128.
\bibitem{sha:open}
L. Shapiro, 
Some open questions about random walks,
involutions, limiting distributions, and
generating functions, {\em Adv. in Appl. Math.} \textbf{27} (2001) 585--596.
\bibitem{bijections}
L. W. Shapiro, Bijections and the Riordan group, {\em 
Theoret. Comput. Sci.} \textbf{307} (2003) 403--413.
\bibitem{ShGeWoWo:Rgroup}
L. W. Shapiro, S. Getu, W. Woan, and L. Woodson, The 
Riordan group, {\em Discrete Appl. Math.}  \textbf{34} (1991) 229--239.
\bibitem{oeis}
N. J. A.~Sloane, \emph{The
On-Line Encyclopedia of Integer Sequences}, \url{https://oeis.org}, 
2018.
\bibitem{sprugnoli} 
R. Sprugnoli,  Riordan arrays and combinatorial sums,
{\em Discrete Math.}  \textbf{132} (1994) 267--290.
\bibitem{yang}
S.-L. Yang, Some inverse relations determined by 
Catalan matrices, {\em Int. J. Comb.} (2013) Art.~ID 528584.
\end{thebibliography}

\bigskip
\hrule
\bigskip

\noindent 2010 {\it Mathematics Subject Classification}:
Primary 05A15; Secondary 05B20, 05A19, 15B36.

\noindent \emph{Keywords: } Riordan group, 
Riordan pseudo-involution, 
automorphism,
infinite array, 
$Z$-sequence, 
$B(x)$-sequence.

\bigskip
\hrule
\bigskip

\noindent 
(Concerned with sequences
\seqnum{A000027},
\seqnum{A000079},
\seqnum{A000124},
\seqnum{A000125},
\seqnum{A000127},
\seqnum{A000217},
\seqnum{A000292},
\seqnum{A000384},
\seqnum{A000389},
\seqnum{A000580},
\seqnum{A000582},
\seqnum{A001288},
\seqnum{A001519},
\seqnum{A002492},
\seqnum{A002662},
\seqnum{A002664},
\seqnum{A005408},
\seqnum{A005843},
\seqnum{A006261},
\seqnum{A008859},
\seqnum{A008860},
\seqnum{A010966},
\seqnum{A010968},
\seqnum{A014105},
\seqnum{A025581},
\seqnum{A035039},
\seqnum{A035041},
\seqnum{A053126},
\seqnum{A053127},
\seqnum{A053128},
\seqnum{A053129},
\seqnum{A059993},
\seqnum{A060163},
\seqnum{A088305},
\seqnum{A114284},
\seqnum{A116722},
\seqnum{A130883},
\seqnum{A145018},
\seqnum{A152948},
\seqnum{A152950},
\seqnum{A165747},
\seqnum{A167499},
\seqnum{A177787},
\seqnum{A201347}, 
and 
\seqnum{A220074}.)

\bigskip
\hrule
\bigskip

\vspace*{+.1in}
\noindent
Received April 2 2019;
revised versions received  November 4 2019; December 24 2019.
Published in {\it Journal of Integer Sequences},
December 26 2019.

\bigskip
\hrule
\bigskip

\noindent
Return to
\htmladdnormallink{Journal of Integer Sequences home page}{https://cs.uwaterloo.ca/journals/JIS/}.
\vskip .1in


\end{document}

</pre></body></html>