Dual-Layer Matrix-Increasing Sequences

• Dual-layer iterative matrix-increasing sequences are defined using two nested iterative processes—the (a,b)-process and a-process—on infinite, invertible, lower-triangular matrices.
• They yield invariant Riordan arrays with closed-form generating functions and explicit coefficients, establishing connections with classical sequences like Catalan and Narayana numbers.
• The framework exposes intricate combinatorial structures through Hankel transforms with Somos-4 recurrences and the generation of eigentriangles that encode linear operator actions.

Dual-layer iterative matrix-increasing sequences denote a structured system of generating, transforming, and interrelating infinite families of combinatorial number triangles by two nested iterative matrix operations, each defined on the space of infinite, invertible, lower-triangular integer matrices with ones along the main diagonal. The framework employs Riordan array formalism to define the operators, characterize their invariants, examine associated Hankel transforms, and introduce the notion of eigentriangles—matrices that recursively encode linear operator actions—resulting in intricate interlocking families of numerical arrays, many with deep combinatorial significance and conjectural connections to Somos-4 type recurrences (Barry, 2011).

1. Iterative Operators on Lower-Triangular Matrices

Two fundamental iterative processes operate on the space of lower-triangular matrices $M=(m_{n,k})_{n, k\ge 0}$ with $m_{n, n}=1$ for all $n$:

• (a,b)-process $\Phi_{a, b}$: Given $M$, a shifted and sign-altered matrix $\widetilde{M}^{(a, b)}$ is constructed:
  • $\tilde m^{(a,b)}_{0,0} = 1$.
  • $\tilde m^{(a,b)}_{n,0} = -a$; $\tilde m^{(a,b)}_{n,1} = -b$ for $n\ge2$.
  • $\tilde m^{(a,b)}_{n,k} = -m_{n-1,k-2}$ for $k\ge2$ and $n\ge k$.

The process updates $M$ as:

$$\Phi_{a, b}(M) = \left( \widetilde{M}^{(a, b)} \right)^{-1}$$

• a-process $\Psi_a$: Analogously, $\widetilde{M}^{(a)}$ is defined by:
  • $\widetilde{M}^{(a)}_{n,0} = -a$.
  • $\widetilde{M}^{(a)}_{n,k} = -m_{n-1, k-1}$ for $k\ge 1$.

This yields:

$$\Psi_a(M) = \left( \widetilde{M}^{(a)} \right)^{-1}$$

Both processes, interpreted in Riordan group terms, transform a general array into a new Appell-type array (of the form $(g(x),x)$).

2. Invariant Arrays and Universal Appell Arrays

Appell (or sequence) Riordan arrays are matrices

$$(g(x), x): \qquad (n, k) \mapsto [k \le n] [x^{n-k}] g(x)$$

where $g(x)$ is a generating function.

• (a,b)-invariant arrays: A Riordan array $(f(x), x)$ is invariant under $\Phi_{a, b}$ if $f(x)$ satisfies

$f(x) = 1 - a x - (b-1)x^2 - x^2 f(x)$

Invariant arrays under $\Phi_{a, b}$ therefore have generating functions as fixed points of this quadratic equation.

• a-invariant arrays: $(g(x), x)$ is invariant under $\Psi_a$ if $g(x)$ solves

$g(x) = 1 - (a-1)x - xg(x)$

Thus, the universal Appell arrays generated by these fixed points form the basis of the "dual-layer" structure.

3. Closed-form Generating Functions and Explicit Coefficient Formulas

The two matrix processes yield universal Appell arrays with explicit generating function and coefficient descriptions:

• (a,b)-Appell arrays: The fixed-point generating function is

$$f(x) = \frac{1 - a x - (b-1)x^2 - \sqrt{(1 - a x - (b-1)x^2)^2 - 4x^2}}{2x^2}$$

with continued fraction expansion and explicit coefficients:

$$a_n = \sum_{k=0}^{\lfloor n/2 \rfloor} \sum_{j=0}^{n-2k} \binom{n-j}{k} \binom{n-2k}{j} (b-1)^j C_k (-1)^j$$

where $C_k$ is the $k$th Catalan number.

• a-Appell (Narayana) arrays: Their generating function is

$$g(x) = \frac{1 - (a-1)x - \sqrt{1 - 2(a+1)x + (a-1)^2 x^2}}{2x}$$

with coefficients given by the Narayana polynomials:

$$N_n(a) = \sum_{k=0}^{n-1} \frac{1}{n} \binom{n}{k} \binom{n}{k+1} a^k$$

These arrays take the form $(f(x), x)$ or $(g(x), x)$, with $(n, k)\mapsto a_{n-k}$ or $N_{n-k}(a)$.

4. Hankel Transforms and Somos-4 Recurrences

For $(a_n)$ defined as the first column of an $(a, b)$-invariant Appell array, the Hankel transform is

$h_n = \det [a_{i+j}]_{0 \leq i, j \leq n}$

It is conjectured that these Hankel transforms obey a Somos-4 type recurrence:

$$h_n = a^2 h_{n-1}h_{n-3} + (b^2 - a^2) h_{n-2}^2, \qquad n\ge4$$

with initial conditions:

$$h_0 = 1,\quad h_1 = b,\quad h_2 = b^2 - a^2,\quad h_3 = b^3 - 3ab^2 + 2a^3$$

For $a=b=1$, this yields $h_n = h_{n-1}h_{n-3} + 3h_{n-2}^2$, i.e., the classical $(1,3)$-Somos-4.

5. Eigentriangles and Eigensequences

Given any invertible lower-triangular matrix $M$, an eigentriangle $E$ is a lower-triangular matrix with first column $(e_n = E_{n,0})$, satisfying

$(ME)_{n,0} = E_{n+1,0},\quad E_{0,0} = 1$

and more generally,

$(ME)_{n,k} = E_{n+1,k}$

The columns of $E$ are recursively generated via:

$$\widetilde E(n, j) = \sum_{k=0}^{n-1} M_{n-1+j,k+j} \widetilde E(k, j),\quad \widetilde E(0,j) = 1$$

with $E(n, k) = \widetilde E(n-k, k)$ for $k\le n$.

Selected Examples

Matrix	Eigentriangle First Column	OEIS Reference
Binomial $B = (\binom{n}{k})$	Bell numbers $B_n$	A000110
Skew-binomial $(\binom{n-k}{k})$	Sequence $1,1,1,2,4,11,33,\dots$	A127782
Motzkin triangle $(M_{n-k})$	Directed animal numbers	A005773

The eigentriangle of the Catalan triangle produces, via $E \cdot (C_1,C_2,\dots)^\top$, the Takeuchi numbers. The corresponding generating function $T(x)$ satisfies Prellberg's functional equation,

$T(x) = \frac{C(x) - 1}{1-x} + x C(x) T(xC(x))$

where $C(x)$ is the Catalan generating function.

6. The Dual-Layer Architecture and Expanding Number Triangle Families

The two matrix processes can be iterated in tandem to form an extensive, hierarchically organized family of number triangles:

• Layer 1 applies the $(a, b)$-process, generating $(f_{a, b}(x), x)$ with conjectural Somos-4 Hankel transforms.
• Layer 2 uses the $a$-process to yield the Narayana-Appell arrays $(g_a(x), x)$.
• Each invariant array admits its own eigentriangle, further expanding the family of integer sequences.
• Iterating the two processes alternately (e.g., $(a_1,b_1)$, then $a_2$) on an initial matrix produces a binary tree of two-parameter families, each governed by nested continued fractions and functional equations.

The generating functions across these families universally satisfy algebraic or quadratic equations of the form

$F(x) = 1 - \alpha x - \beta x^2 - x^k F(x)$

or as continued fractions

$$F(x) = \cfrac{1}{1 - \alpha x - \cfrac{x}{1 - \alpha x - \cdots}}$$

Successive eigentriangles convert these structures into arrays governed by linear recurrences along columns. This structure enables the generation and examination of large, structured families of combinatorial number triangles, with emergent phenomena including novel Hankel transforms and the realization of combinatorial sequences such as Bell, Takeuchi, and directed animal numbers within this layered architecture.