Square Matrices Generated by Sequences of Riordan Arrays

Ângela Mestre and José Agapito

Centro de Análise Funcional, Estruturas Lineares e Aplicações Grupo de Estruturas Algébricas, Lineares e Combinatórias Departamento de Matemática Faculdade de Ciências, Universidade de Lisboa 1749-016 Lisboa Portugal

                   amestre@fc.ul.pt
                    jaruiz@fc.ul.pt

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.

1 Introduction

Let d(t) = 1 + ∑_{k=1}^{∞} d_{k}t^{k} and h(t) = ∑_{k=1}^{∞} h_{k}t^{k} be two generating functions over the field of complex numbers ℂ with h_{1} ≠ 0. A Riordan array is an infinite lower triangular matrix D = (D_{k,n})_{k≥n≥1}, usually represented as a pair D = (d(t), h(t)), whose entries are given by

D_{k,n} = [t^{k−1}]d(t)h(t)^{n−1},  (1)

where [t^{k}] is the usual coefficient extraction operator. The set of Riordan arrays forms a group formally introduced by Shapiro, Getu, Woan, and Woodson [20], here denoted by ℜ𝔦𝔬. The group multiplication is given by

(d_{1}(t), h_{1}(t))(d_{2}(t), h_{2}(t)) = (d_{1}(t)d_{2}(h_{1}(t)), h_{2}(h_{1}(t))),  (2)

while the group identity is id_{ℜ𝔦𝔬} = (1, t). The inverse of any element D = (d(t), h(t)) is D^{−1} = (1/d(h̄(t)), h̄(t)), where h̄(t) denotes the compositional inverse of the generating function h(t), i.e., h(h̄(t)) = h̄(h(t)) = t. There exists an extensive literature on the Riordan group and related concepts as, for instance, [17, 19, 2, 7, 3, 9, 4, 5, 10, 12]. Throughout the years, Riordan arrays have been revealed to be very powerful tools in combinatorics, see, e.g., [22, 15, 16, 11, 13, 8].

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 (D_{n})_{n≥1} be a sequence of Riordan arrays. Fix integers i, j ≥ 1. From each of the arrays in the sequence take the ith column (resp., the jth diagonal or the row-sum vector). Build a square matrix such that the nth column corresponds to the ith column (resp., the jth diagonal or the row-sum vector) of the matrix D_{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 base array of the sequence.

Given α, γ ∈ ℂ and β ∈ ℂ^{∗}, with ℂ^{∗} = ℂ\{0}, Bacher [1] introduced the automorphisms φ_{α,β,γ} : ℜ𝔦𝔬 → ℜ𝔦𝔬 as follows

φ_{α,β,γ} : ℜ𝔦𝔬 → ℜ𝔦𝔬
(d(t), h(t)) ↦ ((\frac{h(t)}{t})^{α}d(t)^{β}h′(t)^{γ}, h(t)) ,  (3)

where h′(t) denotes the derivative of h(t). In particular, the automorphism φ_{1,1,0} is usually known as the diagonal translation operator [1, 14], for φ_{1,1,0}(D) is the matrix obtained from D by removing the first column and the first row (recall formula (1)).

This paper is organized as follows. Given a Riordan array D and fixed complex numbers β ∈ ℂ^{∗} and γ ∈ ℂ, Section 2.1 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 (φ_{1,n,γ}(D))_{n≥1} and (φ_{1,β,l}(D))_{l≥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 3 expresses the A, Z-sequences [15, 9] and the B(x)-sequence [23] of the image of a Riordan array D under an automorphism φ_{α,β,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 (φ_{1,n,γ}(D))_{n≥1}. We illustrate our setting with the classical Pascal array.

Section 4 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 [2] form an array sequence (φ_{1,n,0}(D))_{n≥1}, where D 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 2.1 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 [21] 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.

2 Infinite matrices via Bacher’s automorphisms with nonnegative parameters

This section introduces several infinite square matrices associated with Riordan array sequences through Bacher’s automorphisms parametrized by (1, n, γ) or (1, β, l) with n, l ≥ 1.

Let D = (d(t), h(t)) denote a Riordan array. We restrict the parameter α in the definition of Bacher’s automorphisms (3) to nonnegative integers. Then, for fixed β ∈ ℂ^{∗} and γ ∈ ℂ and nonnegative integers m′ ≥ m, we have φ_{m′,β,γ}(D) = (φ_{m′−m,1,0} ∘ φ_{m,β,γ})(D). 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

ϕ_{β,γ} = φ_{1,β,γ} and ϕ_{n} = φ_{1,n,0}.

Also, as to the notation, note that throughout the paper we shall omit the index 0 associated with γ = 0.

2.1 The array sequence (ϕ_{n,γ}(D))_{n≥1}

Among the sequences that can be associated with a Riordan array D, 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 (ϕ_{n,γ}(D))_{n≥1}.

Definition 1. Fix a number γ ∈ ℂ. For any Riordan array D = (d(t), h(t)) and any integer i ≥ 1, define the following infinite matrices:

(i) Matrix ℭ𝔬𝔩_{i}^{γ,D}, whose nth column is the ith column of ϕ_{n,γ}(D):

(ℭ𝔬𝔩_{i}^{γ,D})_{k,n} = (ϕ_{n,γ}(D))_{k,i} = [t^{k+i−1}]d(t)^{n}h′(t)^{γ}h(t)^{i}, k, n ≥ 1.  (4)

(ii) Matrix 𝔇𝔦𝔞𝔤_{i}^{γ,D}, whose nth column is the ith (descending) diagonal slice of ϕ_{n,γ}(D):

(𝔇𝔦𝔞𝔤_{i}^{γ,D})_{k,n} = (ϕ_{n,γ}(D))_{k+i,k} = [t^{k+i−1}]d(t)^{n}h′(t)^{γ}h(t)^{k}, k, n ≥ 1.  (5)

(iii) Let (𝔯_{n}^{γ,D})_{k}, k ≥ 1, be the sequence whose kth term is the sum of the entries of the kth row of ϕ_{n,γ}(D), for n ≥ 1. In this context, we define the matrix ℜ𝔬𝔴𝔖^{γ,D}, whose nth column is the sequence of the row sums of ϕ_{n,γ}(D):

(ℜ𝔬𝔴𝔖^{γ,D})_{k,n} = (𝔯_{n}^{γ,D})_{k} = [t^{k}]\frac{h(t)d(t)^{n}h′(t)^{γ}}{(1 − h(t))} , k, n ≥ 1.  (6)

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 d(t)/(1 − h(t)) [19, 8].

Note that analogous matrix definitions hold for the sequences (ϕ_{−n,γ}(D))_{n≥1}. In this context, the nth column of the similar matrices, denoted 𝔠𝔬𝔩_{i}^{γ,D}, 𝔡𝔦𝔞𝔤_{i}^{γ,D}, and 𝔯𝔬𝔴𝔖^{γ,D}, corresponds to the ith column, the ith diagonal slice, or the row sum sequence of the matrix ϕ_{−n,γ}(D). Actually, given a Riordan array D, we may even extend the definitions to the sequences ϕ_{±1/n,γ}(D))_{n≥1} so that the nth column of square matrices is associated with a column, diagonal slice or the row sum sequence of the matrix ϕ_{±1/n,γ}(D).

Moreover, to avoid zero rows, in the definition of the square array ℭ𝔬𝔩_{i}^{γ,D}, we removed the first i − 1 zero elements of the ith column of the matrices ϕ_{n,γ}(D) for all i, n ≥ 1. When γ = 0 this definition may be seen as reciprocal to that of Riordan arrays, for the power of the second element of the pair (d(t), h(t)) remains fixed, while the power of the first element increases as the number of columns increases. Indeed, one may even transform the matrix ℭ𝔬𝔩_{i}^{D} 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 ϕ_{i} as given in the next remark. Remark 2. Given any Riordan array D = (d(t), h(t)), let D_{S} = (h(t)/t, td(t)). Then, for any i, n ≥ 1, the nth column of the following Riordan array

ϕ_{i}(D_{S}) = ((\frac{h(t)}{t})^{i} d(t), td(t)) with (ϕ_{i}(D_{S}))_{k,n} = [t^{k+i−n}]d(t)^{n}h(t)^{i}, k, n ≥ 1,

from the nth row on, equals the nth column of the square array ℭ𝔬𝔩_{i}^{D} (formula (4)).

Thus, both (ϕ_{n}(D))_{n≥1} and (ϕ_{n}(D_{S}))_{n≥1} carry the same combinatorial information regarding the columns.

Proposition 3. Let D = (d(t), h(t)) be any Riordan array and let D_{S} = (h(t)/t, td(t)). Then, there is a 1-1 correspondence between the columns of the matrices in the array sequence (ϕ_{n}(D))_{n≥1} and those in the sequence (ϕ_{n}(D_{S}))_{n≥1} such that the ith column of ϕ_{n}(D) is associated with the nth column of ϕ_{i}(D_{S}) for any i, n ≥ 1.

Furthermore, some of the next examples show that the matrices of Definition 1 highlight the shared properties of the matrices in the sequence (ϕ_{n}(D))_{n≥1}.

Example 4. We study the array sequence (ϕ_{−n}(P))_{n≥1}, where P = (1/(1 − t), t/(1 − t)) is the classical Pascal array:

P = [ \begin{matrix}1 & 0 & 0 & 0 & 0 & 0 & 0 & ⋯ \\ 1 & 1 & 0 & 0 & 0 & 0 & 0 & ⋯ \\ 1 & 2 & 1 & 0 & 0 & 0 & 0 & ⋯ \\ 1 & 3 & 3 & 1 & 0 & 0 & 0 & ⋯ \\ 1 & 4 & 6 & 4 & 1 & 0 & 0 & ⋯ \\ 1 & 5 & 10 & 10 & 5 & 1 & 0 & ⋯ \\ 1 & 6 & 15 & 20 & 15 & 6 & 1 & ⋯ \\ ⋮ & ⋮ & ⋮ & ⋮ & ⋮ & ⋮ & ⋮ & ... \end{matrix} ] .  (7)

The first arrays ϕ_{−n}(P) are

ϕ_{−1}(P) = [ \begin{matrix}1 & 0 & 0 & 0 & 0 & 0 & 0 & ⋯ \\ 0 & 1 & 0 & 0 & 0 & 0 & 0 & ⋯ \\ 0 & 1 & 1 & 0 & 0 & 0 & 0 & ⋯ \\ 0 & 1 & 2 & 1 & 0 & 0 & 0 & ⋯ \\ 0 & 1 & 3 & 3 & 1 & 0 & 0 & ⋯ \\ 0 & 1 & 4 & 6 & 4 & 1 & 0 & ⋯ \\ 0 & 1 & 5 & 10 & 10 & 5 & 1 & ⋯ \\ ⋮ & ⋮ & ⋮ & ⋮ & ⋮ & ⋮ & ⋮ & ... \end{matrix} ] ,

ϕ_{−2}(P) = [ \begin{matrix}1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & ⋯ \\ −1 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & ⋯ \\ 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & ⋯ \\ 0 & 0 & 1 & 1 & 0 & 0 & 0 & 0 & ⋯ \\ 0 & 0 & 1 & 2 & 1 & 0 & 0 & 0 & ⋯ \\ 0 & 0 & 1 & 3 & 3 & 1 & 0 & 0 & ⋯ \\ 0 & 0 & 1 & 4 & 6 & 4 & 1 & 0 & ⋯ \\ 0 & 0 & 1 & 5 & 10 & 10 & 5 & 1 & ⋯ \\ ⋮ & ⋮ & ⋮ & ⋮ & ⋮ & ⋮ & ⋮ & ⋮ & ... \end{matrix} ] ,

ϕ_{−3}(P) = [ \begin{matrix}1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & ⋯ \\ −2 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & ⋯ \\ 1 & −1 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & ⋯ \\ 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & ⋯ \\ 0 & 0 & 0 & 1 & 1 & 0 & 0 & 0 & 0 & ⋯ \\ 0 & 0 & 0 & 1 & 2 & 1 & 0 & 0 & 0 & ⋯ \\ 0 & 0 & 0 & 1 & 3 & 3 & 1 & 0 & 0 & ⋯ \\ 0 & 0 & 0 & 1 & 4 & 6 & 4 & 1 & 0 & ⋯ \\ 0 & 0 & 0 & 1 & 5 & 10 & 10 & 5 & 1 & ⋯ \\ ⋮ & ⋮ & ⋮ & ⋮ & ⋮ & ⋮ & ⋮ & ⋮ & ⋮ & ... \end{matrix} ] ,

ϕ_{−4}(P) = [ \begin{matrix}1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & ⋯ \\ −3 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & ⋯ \\ 3 & −2 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & ⋯ \\ −1 & 1 & −1 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & ⋯ \\ 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & ⋯ \\ 0 & 0 & 0 & 0 & 1 & 1 & 0 & 0 & 0 & 0 & ⋯ \\ 0 & 0 & 0 & 0 & 1 & 2 & 1 & 0 & 0 & 0 & ⋯ \\ 0 & 0 & 0 & 0 & 1 & 3 & 3 & 1 & 0 & 0 & ⋯ \\ 0 & 0 & 0 & 0 & 1 & 4 & 6 & 4 & 1 & 0 & ⋯ \\ 0 & 0 & 0 & 0 & 1 & 5 & 10 & 10 & 5 & 1 & ⋯ \\ ⋮ & ⋮ & ⋮ & ⋮ & ⋮ & ⋮ & ⋮ & ⋮ & ⋮ & ⋮ & ... \end{matrix} ] .

Clearly, from the (n + 1)th column on we recover Pascal’s array. We now consider the row sum matrix 𝔯𝔬𝔴𝔖^{P} associated with the sequence (ϕ_{−n}(P))_{n≥1}, whose nth column yields the row sum sequence of the matrix ϕ_{−n}(P):

𝔯𝔬𝔴𝔖^{P} = [ \begin{matrix}1 & 1 & 1 & 1 & 1 & 1 & 1 & ⋯ \\ 1 & 0 & −1 & −2 & −3 & −4 & −5 & ⋯ \\ 2 & 1 & 1 & 2 & 4 & 7 & 11 & ⋯ \\ 4 & 2 & 1 & 0 & −2 & −6 & −13 & ⋯ \\ 8 & 4 & 2 & 1 & 1 & 3 & 9 & ⋯ \\ 16 & 8 & 4 & 2 & 1 & 0 & −3 & ⋯ \\ 32 & 16 & 8 & 4 & 2 & 1 & 1 & ⋯ \\ ⋮ & ⋮ & ⋮ & ⋮ & ⋮ & ⋮ & ⋮ & ... \end{matrix} ] .

We have (𝔯𝔬𝔴𝔖^{P})_{k,n} = ∑_{i=0}^{k−1}(−1)^{i}\binom{n−k+i}{i}. Note that from the (n+1)th element on, the nth column is the row sum sequence of the Pascal array \underline{A000079} whose elements are powers of 2. Now, we examine the upper triangular matrix 𝔯𝔬𝔴^{P} obtained by considering only the n first elements of the nth column of 𝔯𝔬𝔴𝔖^{P} for each n ≥ 1:

𝔯𝔬𝔴^{P} = [ \begin{matrix}1 & 1 & 1 & 1 & 1 & 1 & 1 & ⋯ \\ 0 & 0 & −1 & −2 & −3 & −4 & −5 & ⋯ \\ 0 & 0 & 1 & 2 & 4 & 7 & 11 & ⋯ \\ 0 & 0 & 0 & 0 & −2 & −6 & −13 & ⋯ \\ 0 & 0 & 0 & 0 & 1 & 3 & 9 & ⋯ \\ 0 & 0 & 0 & 0 & 0 & 0 & −3 & ⋯ \\ 0 & 0 & 0 & 0 & 0 & 0 & 1 & ⋯ \\ ⋮ & ⋮ & ⋮ & ⋮ & ⋮ & ⋮ & ⋮ & ... \end{matrix} ] .

Then, the matrix 𝔯𝔬𝔴^{P} is the sequence \underline{A220074} as an upper right triangle.

We now study the array sequences (ϕ_{n}(D))_{n≥1}, where D denotes the triangles \underline{A025581} and \underline{A114284}.

Example 5. We consider the Appell group element T = (p(t)^{2}, t) = (1/(1 − t)^{2}, t) which is the triangle T′ = (tp(t)^{2}, t) \underline{A025581} without the first row. Note that T′ is not in the Riordan group, for the entry (1,1) is zero. The columns of T yield the second column of Pascal’s triangle P = (p(t), tp(t)).

T = [ \begin{matrix}1 & 0 & 0 & 0 & 0 & 0 & ⋯ \\ 2 & 1 & 0 & 0 & 0 & 0 & ⋯ \\ 3 & 2 & 1 & 0 & 0 & 0 & ⋯ \\ 4 & 3 & 2 & 1 & 0 & 0 & ⋯ \\ 5 & 4 & 3 & 2 & 1 & 0 & ⋯ \\ 6 & 5 & 4 & 3 & 2 & 1 & ⋯ \\ ⋮ & ⋮ & ⋮ & ⋮ & ⋮ & ⋮ & ... \end{matrix} ] .

Likewise, the images of T under the automorphisms ϕ_{n} yield the (2n)th columns of the Pascal array P. Thus, the columns of the matrices ℭ𝔬𝔩_{i}^{T} (see formula (4)) are the even columns of Pascal’s triangle for any i ≥ 1:

ℭ𝔬𝔩_{i}^{T} = [ \begin{matrix}1 & 1 & 1 & 1 & 1 & 1 & ⋯ \\ 2 & 4 & 6 & 8 & 10 & 12 & ⋯ \\ 3 & 10 & 21 & 36 & 55 & 78 & ⋯ \\ 4 & 20 & 56 & 120 & 220 & 364 & ⋯ \\ 5 & 35 & 126 & 330 & 715 & 1365 & ⋯ \\ 6 & 56 & 252 & 792 & 2002 & 4368 & ⋯ \\ 7 & 84 & 462 & 1716 & 5005 & 12376 & ⋯ \\ ⋮ & ⋮ & ⋮ & ⋮ & ⋮ & ⋮ & ... \end{matrix} ] .

Indeed, (ℭ𝔬𝔩_{i}^{T})_{k,n} = \binom{2n+k−2}{k−1}, k, n ≥ 1. The first sequence numbers are \underline{A000027}, \underline{A000292}, \underline{A000389}, \underline{A000580}, \underline{A000582}, \underline{A001288}, \underline{A010966}, and \underline{A010968}. From the second row on, the first sequence numbers are \underline{A005843}, \underline{A014105}, \underline{A002492}, \underline{A053126}, \underline{A053127}, and \underline{A053128}.

Example 6. We consider the array sequence (ϕ_{n}(A))_{n≥1}, where A = ((1 − 3t)/(1 − t), t) \underline{A114284}:

A = [ \begin{matrix}1 & 0 & 0 & 0 & 0 & 0 & ⋯ \\ −2 & 1 & 0 & 0 & 0 & 0 & ⋯ \\ −2 & −2 & 1 & 0 & 0 & 0 & ⋯ \\ −2 & −2 & −2 & 1 & 0 & 0 & ⋯ \\ −2 & −2 & −2 & −2 & 1 & 0 & ⋯ \\ −2 & −2 & −2 & −2 & −2 & 1 & ⋯ \\ ⋮ & ⋮ & ⋮ & ⋮ & ⋮ & ⋮ & ... \end{matrix} ] .

The base array A coincides with ϕ_{1}(A). The next arrays ϕ_{n}(A) are

ϕ_{2}(A) = [ \begin{matrix}1 & 0 & 0 & 0 & 0 & 0 & ⋯ \\ −4 & 1 & 0 & 0 & 0 & 0 & ⋯ \\ 0 & −4 & 1 & 0 & 0 & 0 & ⋯ \\ 4 & 0 & −4 & 1 & 0 & 0 & ⋯ \\ 8 & 4 & 0 & −4 & 1 & 0 & ⋯ \\ 12 & 8 & 4 & 0 & −4 & 1 & ⋯ \\ ⋮ & ⋮ & ⋮ & ⋮ & ⋮ & ⋮ & ... \end{matrix} ] ,

ϕ_{3}(A) = [ \begin{matrix}1 & 0 & 0 & 0 & 0 & 0 & ⋯ \\ −6 & 1 & 0 & 0 & 0 & 0 & ⋯ \\ 6 & −6 & 1 & 0 & 0 & 0 & ⋯ \\ 10 & 6 & −6 & 1 & 0 & 0 & ⋯ \\ 6 & 10 & 6 & −6 & 1 & 0 & ⋯ \\ −6 & 6 & 10 & 6 & −6 & 1 & ⋯ \\ ⋮ & ⋮ & ⋮ & ⋮ & ⋮ & ⋮ & ... \end{matrix} ] ,

ϕ_{4}(A) = [ \begin{matrix}1 & 0 & 0 & 0 & 0 & 0 & ⋯ \\ −8 & 1 & 0 & 0 & 0 & 0 & ⋯ \\ 16 & −8 & 1 & 0 & 0 & 0 & ⋯ \\ 8 & 16 & −8 & 1 & 0 & 0 & ⋯ \\ −16 & 8 & 16 & −8 & 1 & 0 & ⋯ \\ −40 & −16 & 8 & 16 & −8 & 1 & ⋯ \\ ⋮ & ⋮ & ⋮ & ⋮ & ⋮ & ⋮ & ... \end{matrix} ] ,

ϕ_{5}(A) = [ \begin{matrix}1 & 0 & 0 & 0 & 0 & 0 & ⋯ \\ −10 & 1 & 0 & 0 & 0 & 0 & ⋯ \\ 30 & −10 & 1 & 0 & 0 & 0 & ⋯ \\ −10 & 30 & −10 & 1 & 0 & 0 & ⋯ \\ −50 & −10 & 30 & −10 & 1 & 0 & ⋯ \\ −42 & −50 & −10 & 30 & −10 & 1 & ⋯ \\ ⋮ & ⋮ & ⋮ & ⋮ & ⋮ & ⋮ & ... \end{matrix} ] .

By formula (6), the matrix for the row sums ℜ𝔬𝔴𝔖^{A} is

ℜ𝔬𝔴𝔖^{A} = [ \begin{matrix}1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & ⋯ \\ −1 & −3 & −5 & −7 & −9 & −11 & −13 & −15 & −17 & ⋯ \\ −3 & −3 & 1 & 9 & 21 & 37 & 57 & 81 & 109 & ⋯ \\ −5 & 1 & 11 & 17 & 11 & −15 & −69 & −159 & −293 & ⋯ \\ −7 & 9 & 17 & 1 & −39 & −87 & −111 & −63 & 121 & ⋯ \\ −9 & 21 & 11 & −39 & −81 & −51 & 99 & 369 & 679 & ⋯ \\ −11 & 37 & −15 & −87 & −51 & 141 & 393 & 465 & 37 & ⋯ \\ −13 & 57 & −69 & −111 & 99 & 393 & 363 & −351 & −1709 & ⋯ \\ −15 & 81 & −159 & −63 & 369 & 465 & −351 & −1791 & −2447 & ⋯ \\ −17 & 109 & −293 & 121 & 679 & 37 & −1709 & −2447 & 479 & ⋯ \\ ⋮ & ⋮ & ⋮ & ⋮ & ⋮ & ⋮ & ⋮ & ⋮ & ⋮ & ... \end{matrix} ] .

Note that the removal of the first line of ℜ𝔬𝔴𝔖^{A} yields a symmetric matrix. Therefore, it follows from formula (3) and the formula for the generating function of the row sum sequence d(t)/(1 − h(t)) [19, 8] that 𝔯_{n}^{A} = (1 − 3t)^{n}/(1 − t)^{n+1} is the generating function for both the nth column and the (n+1)th row of ℜ𝔬𝔴𝔖^{A}, where n ≥ 1. The first column (and the second row) of ℜ𝔬𝔴𝔖^{A} is the odd negative integers sequence \underline{A165747}, i.e., (ℜ𝔬𝔴𝔖^{A})_{k,1} = 1 − 2k, k ≥ 1. The second column (and the third row for n ≥ 3) is the pinwheel numbers sequence \underline{A059993}, i.e., (ℜ𝔬𝔴𝔖^{A})_{k,2} = 2(k − 4)^{2} + 6(k − 4) + 1, k ≥ 4.

2.2 The array sequences (ϕ_{β,l}(D))_{l≥1}

We define infinite square arrays analogous to those of Section 2.1.

Definition 7. Fix a number β ∈ ℂ^{∗}. For any Riordan array D = (d(t), h(t)) and any integer i ≥ 1, define the following infinite matrices:

(i) Matrix 𝔅ℭ𝔬𝔩_{i}^{β,D}, whose lth column is the ith column of ϕ_{β,l}(D):

(𝔅ℭ𝔬𝔩_{i}^{β,D})_{k,l} = (ϕ_{β,l}(D))_{k,i} = [t^{k+i−1}]d(t)^{β}h′(t)^{l}h(t)^{i}, k, l ≥ 1.

(ii) Matrix 𝔅𝔇𝔦𝔞𝔤_{i}^{β,D}, whose lth column is the ith (descending) diagonal slice of ϕ_{β,l}(D):

(𝔅𝔇𝔦𝔞𝔤_{i}^{β,D})_{k,l} = (ϕ_{β,l}(D))_{k+i,k} = [t^{k+i−1}]d(t)^{β}h′(t)^{l}h(t)^{k}, k, l ≥ 1.

(iii) Let (𝔯_{l}^{β,D})_{m}, m ≥ 0, be the sequence whose mth term is the sum of the entries of the mth row of ϕ_{β,l}(D), for l ≥ 1. In this context, we define the matrix 𝔅ℜ𝔬𝔴𝔖^{β,D}, whose lth column is the sequence of the row sums of ϕ_{β,l}(D):

(𝔅ℜ𝔬𝔴𝔖^{β,D})_{k,l} = (𝔯_{l}^{β,D})_{k} = [t^{k}]\frac{h(t)d(t)^{β}h′(t)^{l}}{(1 − h(t))} , k, l ≥ 1.

Moreover, given a Riordan array D, analogous matrix definitions hold for the array sequences (ϕ_{β,−l}(D))_{l≥1} and (ϕ_{β,±1/l}(D))_{l≥1}.

Example 8. We now examine some of the square matrices of Definition 7 taking as base array the Pascal array P = (p(t), tp(t)), with p(t) = 1/(1 − t). First, we fix β = 1 and consider the matrix 𝔅ℭ𝔬𝔩_{i}^{1,P}. We have

(𝔅ℭ𝔬𝔩_{i}^{1,P})_{k,l} = [t^{k−1}]\frac{1}{(1 − t)^{2l+i+1}} , k, l ≥ 1.

Hence, the lth column of 𝔅ℭ𝔬𝔩_{i}^{1,P} is generated by 1/(1 − t)^{2l+i+1} for any l ≥ 1. Therefore,

(𝔅ℭ𝔬𝔩_{i}^{1,P})_{k,l} = \binom{2l + i + k − 1}{2l + i} = \binom{2l + i + k − 1}{k − 1}.

For i = 1 we have

𝔅ℭ𝔬𝔩_{1}^{1,P} = [ \begin{matrix}1 & 1 & 1 & 1 & 1 & 1 & ⋯ \\ 4 & 6 & 8 & 10 & 12 & 14 & ⋯ \\ 10 & 21 & 36 & 55 & 78 & 105 & ⋯ \\ 20 & 56 & 120 & 220 & 364 & 560 & ⋯ \\ 35 & 126 & 330 & 715 & 1365 & 2380 & ⋯ \\ 56 & 252 & 792 & 2002 & 4368 & 8568 & ⋯ \\ 84 & 462 & 1716 & 5005 & 12376 & 27132 & ⋯ \\ 120 & 792 & 3432 & 11440 & 31824 & 77520 & ⋯ \\ ⋮ & ⋮ & ⋮ & ⋮ & ⋮ & ⋮ & ... \end{matrix} ] .

The sequences for the first columns are \underline{A000292}, \underline{A000389}, \underline{A000580}, \underline{A000582}, \underline{A001288}, and \underline{A010966}. From the second row on the sequences for the listed rows are \underline{A005843}, \underline{A014105}, \underline{A002492}, \underline{A053126}, \underline{A053127}, \underline{A053128}, and \underline{A053129}.

Also for β = 1, we have

(𝔅ℜ𝔬𝔴𝔖^{1,P})_{k,l} = [t^{k−1}]\frac{1}{(1 − 2t)(1 − t)^{2l+1}} , k, l ≥ 1.

The lth column of 𝔅ℜ𝔬𝔴𝔖^{1,P} is thus generated by 1/((1 − 2t)(1 − t)^{2l+1}) for any l ≥ 1. Therefore,

(𝔅ℜ𝔬𝔴𝔖^{1,P})_{k,l} = ∑_{i=0}^{k−1} 2^{i}\binom{2l + k − 1 − i}{k − 1 − i} = 2^{k+2l} − ∑_{i=0}^{2l} \binom{k + 2l}{i}.

The matrix yields

𝔅ℜ𝔬𝔴𝔖^{1,P} = [ \begin{matrix}1 & 1 & 1 & 1 & 1 & 1 & ⋯ \\ 5 & 7 & 9 & 11 & 13 & 15 & ⋯ \\ 16 & 29 & 46 & 67 & 92 & 121 & ⋯ \\ 42 & 93 & 176 & 299 & 470 & 697 & ⋯ \\ 99 & 256 & 562 & 1093 & 1941 & 3214 & ⋯ \\ 219 & 638 & 1586 & 3473 & 6885 & 12616 & ⋯ \\ 466 & 1486 & 4096 & 9949 & 21778 & 43796 & ⋯ \\ 968 & 3302 & 26333 & 63004 & 137980 & 280600 & ⋯ \\ ⋮ & ⋮ & ⋮ & ⋮ & ⋮ & ⋮ & ... \end{matrix} ] .

The sequences for the first four columns are \underline{A002662}, \underline{A002664}, \underline{A035039}, and \underline{A035041}. The second and third rows are \underline{A005408} and \underline{A130883}. From the fourth row on and from n such that a(n) = 2^{n} − ∑_{i=0}^{2} \binom{n+2}{i}, n ≥ 4, the sequences for the listed rows yield the alternating elements of the following sequences a(n): \underline{A000125}, \underline{A000127}, \underline{A006261}, \underline{A008859}, and \underline{A008860}.

3 Automorphisms and the A and Z-sequences

Merlini, Sprugnoli, Rogers, and Verri [15], and He and Sprugnoli [9], showed that every Riordan array D = (d(t), h(t)) can be characterized by their A- and Z-sequences. We examine in this Section how the A- and Z-sequences of D are transformed by the automorphisms ϕ_{α,β} = φ_{α,β,0}.

Lemma 9 ([15, 9, 23]). Let D = (d(t), h(t)) be a Riordan array, and let A(t) and Z(t) be the generating functions for the corresponding A- and Z-sequences, respectively. Then,

d(t) = \frac{1}{1 − tZ(h(t))}, h(t) = tA(h(t)).

In terms of the inverse D^{−1} = (1/d(h̄(t)), h̄(t)) we have

A(t) = \frac{t}{h̄(t)} and Z(t) = \frac{A(t)}{t}(1 − \frac{1}{d(h̄(t))}).

Moreover, let (p_{k}(x))_{k≥1} be the polynomial sequence associated with a Riordan array D, which is given by p_{k}(x) = ∑_{n=1}^{k} D_{k,n}x^{n−1}. By the multiplication rule for Riordan arrays (2), the generating function of (p_{k}(x))_{k≥1} is p(t, x) = d(t)/(1 − xh(t)). In this context, if A_{α,β}(t) and Z_{α,β}(t) denote the A- and Z-sequences of ϕ_{α,β}(D), then a simple calculation yields

A_{α,β}(t) = A(t) and Z_{α,β}(t) = \frac{A(t)^{1−α−β}}{t} (A(t)^{α+β} − (A(t) − tZ(t))^{β}).  (8)

Likewise, the generating function p_{α,β}(t, x) of the polynomial sequence of ϕ_{α,β}(D) is given by p_{α,β}(t, x) = (h(t)/t)^{α}d(t)^{β}/(1 − xh(t)). Now, Yang [23] introduced the function

B(t; x) = \frac{A(t) − tZ(t)}{A(t) − xt},

which satisfies

(d(t), h(t))B(t; x) = \frac{1}{1 − xt}.

The function B(t; x) turns out to be the generating function of the polynomial sequence associated with the Riordan array D^{−1}. Note that Yang’s notation omits the variable x in the definition of B(t; x), i.e., B(t; x) = B(t) in [23]. Now, if we define B_{α,β}(t; x) = (A_{α,β}(t) − tZ_{α,β}(t))/(A_{α,β}(t) − xt), this function reads in terms of B(t; x) as follows

B_{α,β}(t; x) = A(t)^{1−α−β}(A(t) − xt)^{β−1}B(t; x)^{β}
= A(t)^{−α}B(t; x)^{β} ∑_{i=0}^{β−1} \binom{β − 1}{i}A(t)^{−i}(−xt)^{i}.  (9)

In particular, for β = 1 we have B_{α,1}(t; x) = A(t)^{−α}B(t; x).

3.1 Infinite matrices associated with the Z and B(x)-sequences

Let D = (d(t), h(t)) be any Riordan array. For each matrix in the sequence (ϕ_{n}(D))_{n≥1}, we denote by Z_{n}^{D} and B_{n}^{D}(x) the corresponding Z and B(x)-sequences. Recall that the A-sequence is invariant under Bacher’s automorphisms. By analogy with Section 2, we define infinite matrices associated with the sequences (Z_{n}^{D}(t))_{n≥1} and (B_{n}^{D}(t; x))_{n≥1}, where the sequences Z_{n}^{D}(t) and B_{n}^{D}(t; x) are defined in terms of the sequences A, Z, and B(t; x) of the matrix D by formulas (8) and (9).

Definition 10. For any Riordan matrix D and any real variable x define the following two infinite matrices:

ℨ_{k,n}^{D} = [t^{k−1}]Z_{n}^{D} and 𝔅_{k,n}^{D}(x) = [t^{k−1}]B_{n}^{D}(x), k, n ≥ 1.  (10)

Hence, by definition the matrices ℨ^{D} and 𝔅^{D}(x) are those whose nth column is generated by the sequences Z and B(x) of ϕ_{n}(D).

Again, analogous matrix definitions 𝔷^{D} and 𝔟^{D}(x) hold for the matrix sequence (ϕ_{−n}(D))_{n≥1}.

Example 11. We study the matrices ℨ^{P} and 𝔅^{P} (x) defined by formula (10), for the Pascal array P = (p(t), tp(t)), where p(t) = 1/(1 − t) (see array (7)). The Pascal triangle has the following well-known A, Z, and B(x)-sequences:

A(t) = 1 + t, Z(t) = 1, and B(t; x) = ∑_{i=0}^{∞}(−1 + x)^{i}t^{i}.

The matrix ℨ^{P} yields

ℨ^{P} = [ \begin{matrix}2 & 3 & 4 & 5 & 6 & 7 & ⋯ \\ −1 & −3 & −6 & −10 & −15 & −21 & ⋯ \\ 1 & 4 & 10 & 20 & 35 & 56 & ⋯ \\ −1 & −5 & −15 & −35 & −70 & −126 & ⋯ \\ 1 & 6 & 21 & 56 & 126 & 252 & ⋯ \\ −1 & −7 & −28 & −84 & 210 & −462 & ⋯ \\ 1 & 8 & 36 & 120 & 330 & 792 & ⋯ \\ −1 & −9 & −45 & −165 & −495 & −1287 & ⋯ \\ ⋮ & ⋮ & ⋮ & ⋮ & ⋮ & ⋮ & ... \end{matrix} ] .

Then, by taking absolute values for the entries, the rows of ℨ^{P} are the columns of the matrix ϕ_{1}(P). More precisely,

ℨ_{k,n}^{P} = (−1)^{k−1}\binom{n + k − 1}{k}, n ≥ 1, k > 1.

Next, we consider the matrix 𝔅^{P} (x):

𝔅^{P} (x) = [ \begin{matrix}1 & 1 & 1 & ⋯ \\ −2 + x & −3 + x & −4 + x & ⋯ \\ 3 − 3x + x^{2} & 6 − 4x + x^{2} & 10 − 5x + x^{2} & ⋯ \\ −4 + 6x − 4x^{2} + x^{3} & −10 + 10x − 5x^{2} + x^{3} & −20 + 15x − 6x^{2} + x^{3} & ⋯ \\ 5 − 10x + 10x^{2} − 5x^{3} + x^{4} & 15 − 20x + 15x^{2} − 6x^{3} + x^{4} & 35 − 35x + 21x^{2} − 7x^{3} + x^{4} & ⋯ \\ ⋮ & ⋮ & ⋮ & ... \end{matrix} ] .

The entries 𝔅^{P} (x) are thus monic polynomials of degree k − 1:

𝔅_{k,n}^{P}(x) = ∑_{i=0}^{k−1}(−1)^{k−1+i}\binom{n + k − 1}{n + i}x^{i}, k, n ≥ 1.

For x = 1, the matrix 𝔅^{P} (1) is Pascal’s array (read as a square matrix) with alternating signs:

𝔅^{P} (1) = [ \begin{matrix}1 & 1 & 1 & 1 & 1 & 1 & ⋯ \\ −1 & −2 & −3 & −4 & −5 & −6 & ⋯ \\ 1 & 3 & 6 & 10 & 15 & 21 & ⋯ \\ −1 & −4 & −10 & −20 & −35 & −56 & ⋯ \\ 1 & 5 & 15 & 35 & 70 & 126 & ⋯ \\ −1 & −6 & −21 & −56 & −126 & −462 & ⋯ \\ 1 & 7 & 28 & 84 & 330 & 792 & ⋯ \\ −1 & −8 & −36 & −120 & −495 & −1287 & ⋯ \\ ⋮ & ⋮ & ⋮ & ⋮ & ⋮ & ⋮ & ... \end{matrix} ] .

4 Automorphisms and Riordan group pseudo-involutions

We study the action of Bacher’s automorphisms on the pseudo-involutions of the Riordan group.

We recall that a Riordan array D = (d(t), h(t)) is an involution if and only if

d(t) = \frac{1}{d(h(t))},
h(h(t)) = t.

Note that for any involution I and any Riordan array D, the identity (DI)^{2} = id_{ℜ𝔦𝔬} is equivalent to (ID)^{2} = id_{ℜ𝔦𝔬}. Therefore, since D has an inverse, we have (DI)^{2} = id_{ℜ𝔦𝔬} if and only if (D^{n}I)^{2} = id_{ℜ𝔦𝔬} for any integer n. In this context, Shapiro [18] and later Cameron and Nkwanta [2] introduced the following notion of pseudo-involution. An element D of the Riordan group has pseudo-order 2 if the matrix DM has order 2, where M = (1, −t) is an involution. Therefore, a Riordan array D = (d(t), h(t)) with pseudo-order 2 is known as a pseudo-involution. In this case, it readily follows that

d(t) = \frac{1}{d(−h(t))},  (11)
h(−h(t)) = −t.  (12)

In particular, if D = (1, h(t)) is an element of the Lagrange subgroup such that

h(t) = h̄(t) and h(−t) = −h(t),

then D has pseudo-order 2 [2]. We next give the following characterization of pseudoinvolutions.

Proposition 12. Let α : ℜ𝔦𝔬 → ℜ𝔦𝔬 be any Riordan group automorphism. If α(M) = M, then an element D ∈ ℜ𝔦𝔬 has pseudo-order 2 if and only if α(D) has pseudo-order 2.

Proof. Suppose that DM has order 2. Then, (α(D)M)^{2} = α((DM)^{2}) = α(id_{ℜ𝔦𝔬}) = id_{ℜ𝔦𝔬}. Suppose now that α(D) has pseudo-order 2. Then, id_{ℜ𝔦𝔬} = (α(D)M)^{2} = α((DM)^{2}). Therefore, (DM)^{2} = id_{ℜ𝔦𝔬}, for α is injective.

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 φ_{α,β,γ}(M) = ((−1)^{α+γ}, −t), not all the automorphisms φ_{α,β,γ} leave the Riordan array M invariant. Nevertheless, they all preserve pseudo-involutions as we show next.

Proposition 13. For any α, γ ∈ ℂ and β ∈ ℂ^{∗}, a Riordan matrix D ∈ ℜ𝔦𝔬 has pseudo-order 2 if and only if φ_{α,β,γ}(D) has pseudo-order 2.

Proof. Suppose that D = (d(t), h(t)) has pseudo-order 2. By equations (11) and (12) we have

(φ_{α,β,γ}(DM)^{2} = ((\frac{h(t)}{t})^{α}d(t)^{β}h′(t)^{γ}, −h(t))^{2}
= ((\frac{h(t)}{t})^{α}d(t)^{β}h′(t)^{γ}(\frac{h(−h(t))}{−h(t)})^{α}d(−h(t))^{β}h′(−h(t))^{γ}, t)
= (1, t).

Now, suppose that ϕ_{r,s}(D) has pseudo-order 2. From equation (11) we have

(\frac{h(t)}{t})^{α}d(t)^{β}h′(t)^{γ} = (\frac{−h(t)}{h(−h(t))})^{α} \frac{1}{d(−h(t))^{β}h′(−h(t))^{γ}} .

Using h(−h(t)) = −t from equation (12) gives d(t) = 1/d(−h(t)).

This proposition extends Cheon, Kim, Jin, and Shapiro’s remark that says that if (h(t)/t, h(t)) is a pseudo-involution, then ((h(t)/t)^{n}, h(t)) is also a pseudo-involution for all integers n [4].

Example 14. Consider the generalized RNA triangles introduced by Cameron and Nkwanta [2] for n ≥ 1:

D_{±n} = (g (t) (\frac{1 − t}{1 − tg (t)})^{±n} , tg (t)) ,

where

g (t) = \frac{1 − t + t^{2} − \sqrt{1 − 2t − t^{2} − 2t^{3} + t^{4}}}{2t^{2}} .

We recall that D_{±n} = A^{∓n}D_{0}A^{±n}, where D_{0} = (g (t) , tg (t)) is the Nkwanta RNA triangle, and A^{±n} = (\frac{1}{(1−t)^{±n}} , t). We now give an alternative definition of the matrices D_{±n} in terms of the automorphisms ϕ_{±n} for n ≥ 1. Let

D = (\frac{1 − t}{1 − tg(t)}, tg(t)) ,  (13)

D = [ \begin{matrix}1 & 0 & 0 & 0 & 0 & 0 & ⋯ \\ 0 & 1 & 0 & 0 & 0 & 0 & ⋯ \\ 1 & 1 & 1 & 0 & 0 & 0 & ⋯ \\ 2 & 2 & 2 & 1 & 0 & 0 & ⋯ \\ 5 & 5 & 4 & 3 & 1 & 0 & ⋯ \\ 12 & 12 & 10 & 7 & 4 & 1 & ⋯ \\ ⋮ & ⋮ & ⋮ & ⋮ & ⋮ & ⋮ & ... \end{matrix} ] .

The diagonal slices of the array D are of combinatorial interest. The sequence numbers for the first three nontrivial diagonals are \underline{A000124}, \underline{A177787}, and \underline{A116722}. Moreover, the matrix D given by formula (13) is the base matrix for the sequences (D_{±n})_{n≥1}. That is, for n ≥ 1 we have

D_{±n} = ϕ_{±n}(D).

Also, the matrices ϕ_{α,β}(D_{±n}) can be expressed in terms of the base matrix D as follows

ϕ_{α,β}(D_{±n}) = (ϕ_{α,β} ∘ ϕ_{±n})(D) = ϕ_{α+β,(±n)β}(D).

Note that the matrix D is not a pseudo-involution. Indeed,

(DM)^{2} = (\frac{1 − t}{1 − tg(t)}, −tg(t))^{2} = (\frac{1 − t}{1 − tg(t)} \frac{1 + tg(t)}{1 + t}, t) ≠ (1, t),

since g(t) satisfies the functional equation g(t)g(−tg(t)) = 1. Therefore, it follows directly from Proposition 13 that the matrices D_{±n} are not pseudo-involutions either, which contradicts Theorem 3 of [2]. Observe also that although the aforementioned definition of the generalized RNA matrices, namely,

D_{±n} = A^{∓n}D_{0}A^{±n} = ((1 − t)^{±n}, t)(g(t), tg(t)) (\frac{1}{(1 − t)^{n}} , t)

given in [2] does not yield pseudo-involutions, we can easily define pseudo-involutions from Nkwanta’s RNA matrix by setting

D_{±n}^{±} = ((1 ± t)^{±n}, t)D_{0} (\frac{1}{(1 ∓ t)^{±n}} , t) = (g(t)\frac{1 ± t}{1 ∓ tg(t)}, tg(t)), n ≥ 1.

It is worth examining some of the matrices ℭ𝔬𝔩_{i}^{D}, 𝔇𝔦𝔞𝔤_{i}^{D}, and ℜ𝔬𝔴𝔖^{D} (recall formulas (4), (5), and (6) of Section 2.1). First, we study the matrix ℭ𝔬𝔩_{1}^{D} whose columns are given by the first column of each of the matrices D_{n} for n ≥ 1. We have

ℭ𝔬𝔩_{1}^{D} = [ \begin{matrix}1 & 1 & 1 & 1 & 1 & 1 & ⋯ \\ 1 & 1 & 1 & 1 & 1 & 1 & ⋯ \\ 2 & 3 & 4 & 5 & 6 & 7 & ⋯ \\ 5 & 8 & 11 & 14 & 17 & 20 & ⋯ \\ 12 & 21 & 31 & 42 & 54 & 67 & ⋯ \\ 29 & 55 & 86 & 122 & 163 & 209 & ⋯ \\ 71 & 144 & 237 & 351 & 487 & 646 & ⋯ \\ 175 & 377 & 650 & 1001 & 1437 & 1965 & ⋯ \\ 434 & 987 & 1775 & 2833 & 4197 & 5904 & ⋯ \\ ⋮ & ⋮ & ⋮ & ⋮ & ⋮ & ⋮ & ... \end{matrix} ] .

The second column is the sequence \underline{A088305}. Moreover, the sequences for the first nontrivial rows are • (ℭ𝔬𝔩_{1}^{D})_{4,n} = 2 + 3n, n ≥ 1;

• (ℭ𝔬𝔩_{1}^{D})_{5,1} = 12 and (ℭ𝔬𝔩_{1}^{D})_{5,n} = (ℭ𝔬𝔩_{1}^{D})_{5,n−1} + 7 + n, n > 1;

• (ℭ𝔬𝔩_{1}^{D})_{6,1} = 29 and (ℭ𝔬𝔩_{1}^{D})_{6,n} = (ℭ𝔬𝔩_{1}^{D})_{6,n−1} + 16 + 5n, n > 1;

• (ℭ𝔬𝔩_{1}^{D})_{7,1} = 71, (ℭ𝔬𝔩_{1}^{D})_{7,2} = 144, and (ℭ𝔬𝔩_{1}^{D})_{7,n} = 2(ℭ𝔬𝔩_{1}^{D})_{7,n−1} − (ℭ𝔬𝔩_{1}^{D})_{7,n−2} + 17 + n, n > 2;

• (ℭ𝔬𝔩_{1}^{D})_{8,1} = 175, (ℭ𝔬𝔩_{1}^{D})_{8,2} = 377, and (ℭ𝔬𝔩_{1}^{D})_{8,n} = 2(ℭ𝔬𝔩_{1}^{D})_{8,n−1} − (ℭ𝔬𝔩_{1}^{D})_{8,n−2} + 50 + 7n, n > 2;

• (ℭ𝔬𝔩_{1}^{D})_{9,1} = 434, (ℭ𝔬𝔩_{1}^{D})_{9,2} = 987, and (ℭ𝔬𝔩_{1}^{D})_{9,n} = 2(ℭ𝔬𝔩_{1}^{D})_{9,n−1} − (ℭ𝔬𝔩_{1}^{D})_{9,n−2} + 136 +

(n^{2} + 63n)/2, n > 2. We now analyse the first nontrivial diagonal slices of the matrices D_{n}. The matrix of the third diagonal slices reads as follows

𝔇𝔦𝔞𝔤_{3}^{D} = [ \begin{matrix}2 & 3 & 4 & 5 & 6 & 7 & ⋯ \\ 4 & 5 & 6 & 7 & 8 & 9 & ⋯ \\ 7 & 8 & 9 & 10 & 11 & 12 & ⋯ \\ 11 & 12 & 13 & 14 & 15 & 16 & ⋯ \\ 16 & 17 & 18 & 19 & 20 & 21 & ⋯ \\ 22 & 23 & 24 & 25 & 26 & 27 & ⋯ \\ 29 & 30 & 31 & 32 & 33 & 34 & ⋯ \\ 37 & 38 & 39 & 40 & 41 & 42 & ⋯ \\ ⋮ & ⋮ & ⋮ & ⋮ & ⋮ & ⋮ & ... \end{matrix} ] .

The entries are given by

(𝔇𝔦𝔞𝔤_{3}^{D})_{k,n} = n + 1 + \frac{(k + 2)^{2} − 3(k + 2)}{2}, k, n ≥ 1.

Also, the following recursive definition holds

(𝔇𝔦𝔞𝔤_{3}^{D})_{1,n} = n + 1,
(𝔇𝔦𝔞𝔤_{3}^{D})_{k,n} = (𝔇𝔦𝔞𝔤_{3}^{D})_{k−1,n} + k − 1, k > 1.

The first five sequence numbers are \underline{A000124}, \underline{A152948}, \underline{A152950}, \underline{A145018}, and \underline{A167499}. Furthermore, as for the fourth diagonal slices of the matrices D_{n} we have

𝔇𝔦𝔞𝔤_{4}^{D} = [ \begin{matrix}5 & 8 & 11 & 14 & 17 & 20 & ⋯ \\ 10 & 14 & 18 & 22 & 26 & 30 & ⋯ \\ 18 & 23 & 28 & 33 & 38 & 43 & ⋯ \\ 30 & 36 & 42 & 48 & 54 & 60 & ⋯ \\ 47 & 54 & 61 & 68 & 75 & 82 & ⋯ \\ 70 & 78 & 86 & 94 & 102 & 110 & ⋯ \\ 100 & 109 & 118 & 127 & 136 & 145 & ⋯ \\ 138 & 148 & 158 & 168 & 178 & 188 & ⋯ \\ ⋮ & ⋮ & ⋮ & ⋮ & ⋮ & ⋮ & ... \end{matrix} ] .

The first column is of course the fourth diagonal slice of ϕ_{1}(D) (and of the base matrix D), sequence \underline{A177787}. The second column is the sequence \underline{A201347}. The entries of 𝔇𝔦𝔞𝔤_{4}^{D} are given by

(𝔇𝔦𝔞𝔤_{4}^{D})_{k,n} = \frac{(k + 1)(11 + (k + 1)^{2})}{2} + (2 + k)(n − 1), k, n ≥ 1.

Equivalently, the sequence for the kth row, k ≥ 1, yields

(𝔇𝔦𝔞𝔤_{4}^{D})_{k,n} = (𝔇𝔦𝔞𝔤_{4}^{D})_{k,n−1} + k + 2, n > 1.

The recursion formula for the nth column, n ≥ 1, is

(𝔇𝔦𝔞𝔤_{4}^{D})_{1,1} = 5;
(𝔇𝔦𝔞𝔤_{4}^{D})_{1,n} = (𝔇𝔦𝔞𝔤_{4}^{D})_{1,n−1} + 3, n > 1,
(𝔇𝔦𝔞𝔤_{4}^{D})_{k,n} = (𝔇𝔦𝔞𝔤_{4}^{D})_{k−1,n} + n + k + 6, k > 1.

We next study the matrix ℜ𝔬𝔴𝔖^{D} for the row sum sequences of the matrices D_{n}:

ℜ𝔬𝔴𝔖^{D} = [ \begin{matrix}1 & 1 & 1 & 1 & 1 & 1 & ⋯ \\ 2 & 2 & 2 & 2 & 2 & 2 & ⋯ \\ 5 & 6 & 7 & 8 & 9 & 10 & ⋯ \\ 13 & 17 & 21 & 25 & 29 & 33 & ⋯ \\ 34 & 48 & 63 & 79 & 96 & 114 & ⋯ \\ 89 & 134 & 185 & 242 & 305 & 374 & ⋯ \\ 233 & 371 & 536 & 729 & 951 & 1203 & ⋯ \\ 610 & 1021 & 1537 & 2166 & 2916 & 3795 & ⋯ \\ ⋮ & ⋮ & ⋮ & ⋮ & ⋮ & ⋮ & ... \end{matrix} ] .

The first column is the sequence \underline{A001519}:

(ℜ𝔬𝔴𝔖^{D})_{1,1} = 1,
(ℜ𝔬𝔴𝔖^{D})_{2,1} = 1,
(ℜ𝔬𝔴𝔖^{D})_{k,n} = 3(ℜ𝔬𝔴𝔖^{D})_{k−1,1} − (ℜ𝔬𝔴𝔖^{D})_{k−2,1}, k > 1.

Also, the sequences for the first nontrivial rows are • (ℜ𝔬𝔴𝔖^{D})_{4,n} = 9 + 4n, n ≥ 1;

• (ℜ𝔬𝔴𝔖^{D})_{5,1} = 34 and (ℜ𝔬𝔴𝔖^{D})_{5,n} = (ℜ𝔬𝔴𝔖^{D})_{5,n−1} + 12 + n, n > 1;

• (ℜ𝔬𝔴𝔖^{D})_{6,1} = 89 and (ℜ𝔬𝔴𝔖^{D})_{6,n} = (ℜ𝔬𝔴𝔖^{D})_{6,n−1} + 33 + 6n, n > 1;

• (ℜ𝔬𝔴𝔖^{D})_{7,1} = 233, (ℜ𝔬𝔴𝔖^{D})_{8,2} = 371, and (ℜ𝔬𝔴𝔖^{D})_{7,n} = 2(ℜ𝔬𝔴𝔖^{D})_{7,n−1} − (ℜ𝔬𝔴𝔖^{D})_{7,n−2} + 24 + n, n > 2;

• (ℜ𝔬𝔴𝔖^{D})_{8,1} = 610, (ℜ𝔬𝔴𝔖^{D})_{8,2} = 1021, and (ℜ𝔬𝔴𝔖^{D})_{8,n} = 2(ℜ𝔬𝔴𝔖^{D})_{8,n−1} − (ℜ𝔬𝔴𝔖^{D})_{8,n−2} + 81 + 8n, n > 2.

Example 15. We study some of the matrices for the diagonal slices of Section 2.2 taking as base the matrix D_{0} = (g(t), tg(t)). For fixed β ∈ ℂ^{∗}, the sequences (D_{±l}^{β})_{l≥1} = (ϕ_{β,±l}(D_{0}))_{l≥1} are sequences of pseudo-involutions by Proposition 13. For the matrix D_{0}, the second formula of Definition 7 yields

(𝔅𝔇𝔦𝔞𝔤_{i}^{β,D_{0}})_{k,l} = [t^{i−1}]((tg)′(t))^{l}g(t)^{k+β}, k, l ≥ 1.

Hence, restricting β to integers n ≠ 0, the matrices 𝔅𝔇𝔦𝔞𝔤_{i}^{n+1,D_{0}} are obtained from 𝔅𝔇𝔦𝔞𝔤_{i}^{n,D_{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 β = 1. The value i = 2 defines the first nontrivial square matrix:

𝔅𝔇𝔦𝔞𝔤_{2}^{1,D_{0}} = [ \begin{matrix}4 & 6 & 8 & 10 & 12 & 14 & ⋯ \\ 5 & 7 & 9 & 11 & 13 & 15 & ⋯ \\ 6 & 8 & 10 & 12 & 14 & 16 & ⋯ \\ 7 & 9 & 11 & 13 & 15 & 17 & ⋯ \\ 8 & 10 & 12 & 14 & 16 & 18 & ⋯ \\ 9 & 11 & 13 & 15 & 17 & 19 & ⋯ \\ 10 & 12 & 14 & 16 & 18 & 20 & ⋯ \\ 11 & 13 & 15 & 17 & 19 & 21 & ⋯ \\ ⋮ & ⋮ & ⋮ & ⋮ & ⋮ & ⋮ & ... \end{matrix} ] .

Hence, we have

(𝔅𝔇𝔦𝔞𝔤_{2}^{1,D_{0}})_{k,l} = k + 2l + 1, k, l ≥ 1.

Now, for i = 3 the matrix for the diagonal slices of D_{l}^{1} yields

𝔅𝔇𝔦𝔞𝔤_{3}^{1,D_{0}} = [ \begin{matrix}10 & 21 & 36 & 55 & 78 & 105 & ⋯ \\ 15 & 28 & 45 & 66 & 91 & 120 & ⋯ \\ 21 & 36 & 55 & 78 & 105 & 136 & ⋯ \\ 28 & 45 & 66 & 91 & 120 & 153 & ⋯ \\ 36 & 55 & 78 & 105 & 136 & 171 & ⋯ \\ 45 & 66 & 91 & 120 & 153 & 190 & ⋯ \\ 55 & 78 & 105 & 136 & 171 & 210 & ⋯ \\ 66 & 91 & 120 & 153 & 190 & 231 & ⋯ \\ ⋮ & ⋮ & ⋮ & ⋮ & ⋮ & ⋮ & ... \end{matrix} ] .

The sequence for the odd rows is \underline{A014105}, while that for the even rows is \underline{A000384}. As for the columns, the sequence is \underline{A000217}. We have

(𝔅𝔇𝔦𝔞𝔤_{3}^{1,D_{0}})_{k,l} = \binom{k + 2l + 2}{2}, k, l ≥ 1.

The next matrix for the diagonal slices of D_{l}^{1} is

𝔅𝔇𝔦𝔞𝔤_{4}^{1,D_{0}} = [ \begin{matrix}26 & 66 & 134 & 238 & 386 & 586 & ⋯ \\ 42 & 95 & 180 & 305 & 478 & 707 & ⋯ \\ 64 & 132 & 236 & 384 & 584 & 844 & ⋯ \\ 93 & 178 & 303 & 476 & 705 & 998 & ⋯ \\ 130 & 234 & 382 & 582 & 842 & 1170 & ⋯ \\ 176 & 301 & 474 & 703 & 996 & 1361 & ⋯ \\ 232 & 380 & 580 & 840 & 1168 & 1572 & ⋯ \\ 299 & 472 & 701 & 994 & 1359 & 1804 & ⋯ \\ ⋮ & ⋮ & ⋮ & ⋮ & ⋮ & ⋮ & ... \end{matrix} ] .

The sequences for the first two columns are \underline{A000125} and \underline{A060163}. The equation for the entries is

(𝔅𝔇𝔦𝔞𝔤_{4}^{1,D_{0}})_{k,l} = \frac{(k + 2(l + 1))^{3} + 5(k + 2(l + 1)) + 6(2l − 1)}{6}, k, l ≥ 1.

5 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).

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2010 Mathematics Subject Classification: Primary 05A15; Secondary 05B20, 05A19, 15B36. Keywords: Riordan group, Riordan pseudo-involution, automorphism, infinite array, Zsequence, B(x)-sequence.

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

Received April 2 2019; revised versions received November 4 2019; December 24 2019. Published in Journal of Integer Sequences, December 26 2019.

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