THE GENERATING FUNCTION OF THE CATALAN NUMBERS AND LOWER TRIANGULAR INTEGER MATRICES

In the paper, by the Faà di Bruno formula, several identities for the Bell polynomials of the second kind, and an inversion theorem, the authors simplify coefficients of two families of nonlinear ordinary differential equations for the generating function of the Catalan numbers and discover inverses of fifteen closely related lower triangular integer matrices. 1. Motivation The Catalan numbers Cn for n ≥ 0 form a combinatorial sequence of natural numbers that occur in tree enumeration problems such as “In how many ways can a regular n-gon be divided into n− 2 triangles if different orientations are counted E-mail addresses: qifeng618@gmail.com, qifeng618@hotmail.com, qxlxajh@163.com, yaoyonghong@aliyun.com. 2010 Mathematics Subject Classification. Primary 05A15; Secondary 11B65, 11B75, 11B83, 15A09, 15B36, 34A05, 34A34, 40E99.

In [4,Theorem 2.1], Kims established recursively and inductively that the family of differential equations In [4, Theorem 3.1], by similar argument as in the proof of [4, Theorem 2.1], they found that the family of differential equations , where t denotes the floor function whose value is the largest integer less than or equal to t, the coefficients b 0 (n) = 1 and In [4, Theorems 2.2 and 3.2] and [4,Remark], they also used the coefficients a i (n) and b i (n) respectively defined in (1.3) and (1.5) to express their other results in [4].In other words, the quantities a i (n) and b i (n) are the core of the paper [4].
It is obvious that the coefficients a i (n) and b i (n) respectively defined in (1.3) and (1.5) can not be easily remembered, possibly understood, and simply computed.
The aim of this paper is the same one as in the papers [11,14,17,18,24,25,26,27,30,31,32,38,39,40,41,45] and closely related references therein.Concretely speaking, our aim in this paper is to discover simple, significant, meaningful, easily remembered, possibly understood, readily computed expressions for the coefficients a i (n) and b i (n) in the families (1.2) and (1.4) respectively.Consequently and equivalently, we further derive inverses of fifteen closely related lower triangular integer matrices.

Lemmas
To reach our aim in this paper, we recall the following lemmas.
Lemma 2.1 ([2, pp.134 and 139]).The Faà di Bruno formula can be described in terms of the Bell polynomials of the second kind B n,k (x 1 , x 2 , . . ., x n−k+1 ) by for n ≥ 0, where the Bell polynomials of the second kind, or say, partial Bell polynomials, denoted by B n,k (x 1 , x 2 , . . ., x n−k+1 ) for n ≥ k ≥ 0, are defined by where a and b are any complex numbers.

Main results and their proofs
Now we are in a position to state our main results and to prove them simply.
Theorem 3.1.For n ∈ N, the nth derivative and the powers of the generating function G(x) defined in (1.1) satisfy for n ∈ N. Further making use of the formula (2.3) and simplifying arrive at The proof of Theorem 3.1 is complete.
This expression is quite simpler, more easily remembered, more possibly understood, more readily computed, more significant, and more meaningful than the one in (1. Proof.The derivative formula (3.1) can be rearranged as Considering Lemma 2.4 leads straightforwardly to The proof of Theorem 3.2 is complete.
This expression is rather simpler, more easily remembered, more possibly understood, more readily computed, more significant, and more meaningful than the one in (1.5)!!!

Inverses of lower triangular integer matrices
Every inversion theorem in combinatorics corresponds to a lower triangular invertible matrix and its inverse.Conversely, every lower triangular invertible matrix and its inverse correspond to an inversion theorem.Generally, it is not easy to compute the inverse of a lower triangular invertible matrix.
Lemma 2.4 is equivalent to that the lower triangular integer matrices A n = a i,j 1≤i,j≤n and B n = b i,j 1≤i,j≤n with are inversive to each other.See [14, Thorem 8.1] and [41, Theorem 4.1] also.
Theorem 4.1.For n ∈ N, the lower triangular integer matrices P n = p i,j 1≤i,j≤n and Q n = q i,j 1≤i,j≤n with Proof.This follows from rewriting the equations (3.1) and (3.2) as and This also follows from rearranging Lemma 2.4 as The proof of Theorem 4.1 is complete.
By similar argument as in the proof of Theorem 4.1, we derive the following conclusions about lower triangular integer matrices and their inverses.Theorem 4.2.For n ∈ N, the lower triangular integer matrices U n = u i,j 1≤i,j≤n and V n = v i,j 1≤i,j≤n with are inversive to each other.
Theorem 4.3.For n ∈ N, the lower triangular integer matrices C n = c i,j 1≤i,j≤n and D n = d i,j 1≤i,j≤n with and are inversive to each other.For n ∈ N, the lower triangular integer matrices Y n = y i,j 1≤i,j≤n and Z n = z i,j 1≤i,j≤n with and are inversive to each other.
Theorem 4.5.For n ∈ N, the lower triangular integer matrices E n = e i,j 1≤i,j≤n and F n = f i,j 1≤i,j≤n with and are inversive to each other.
Theorem 4.6.For n ∈ N, the lower triangular integer matrices G n = g i,j 1≤i,j≤n and H n = h i,j 1≤i,j≤n with and are inversive to each other.
Theorem 4.7.For n ∈ N, the lower triangular integer matrices K n = k i,j 1≤i,j≤n and L n = i,j 1≤i,j≤n with are inversive to each other.
Theorem 4.8.For n ∈ N, the lower triangular integer matrices M n = m i,j 1≤i,j≤n and N n = n i,j 1≤i,j≤n with and are inversive to each other.
Theorem 4.9.For n ∈ N, the lower triangular integer matrices O n = o i,j 1≤i,j≤n and R n = r i,j 1≤i,j≤n with are inversive to each other.
Theorem 4.10.For n ∈ N, the lower triangular integer matrices T n = t i,j 1≤i,j≤n and W n = w i,j 1≤i,j≤n with are inversive to each other.
Theorem 4.11.For n ∈ N, the lower triangular integer matrices Λ n = λ i,j 1≤i,j≤n and Θ n = θ i,j 1≤i,j≤n with are inversive to each other.
Theorem 4.12.For n ∈ N, the lower triangular integer matrices Φ n = φ i,j 1≤i,j≤n and Ψ n = ψ i,j 1≤i,j≤n with are inversive to each other.Theorem 4.14.For n ∈ N, the lower triangular integer matrices P n = µ i,j 1≤i,j≤n and Q n = ν i,j 1≤i,j≤n with are inversive to each other.

. 1 )√ 1 −
Proof.This proof is a slight modification of the first part in the second proof of[35,  Theorem 1.1].Taking f (u) = 2 1+u and u = h(x) = 4x in the formula (2.1) and utilizing the identity (2.2) yield