Simplifying coefficients in differential equations for generating function of Catalan numbers

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 in two families of nonlinear ordinary differential equations for the generating function of the Catalan numbers.

In [3, Theorems 2.2 and 3.2, [3,Remark]], Kim and Kim also used the coefficients a i (n) and b i (n) respectively defined in (3) and (5) to express their other results in [3]. In other words, the quantities a i (n) and b i (n) are the core of the paper [3]. It is obvious that the coefficients a i (n) and b i (n) respectively defined in (3) and (5) can not be easily remembered, possibly understood, and simply computed.

Lemmas
To reach our aim in this paper, we recall the following lemmas.

Lemma 2.1 ([19, p. 134 and 139]): The Faà di Bruno formula can be described in terms of the Bell polynomials of the second kind
for n ≥ 0, where the Bell polynomials of the second kind, or say, partial Bell polynomials, denoted by where a and b are any complex numbers.
where the double factorial of negative odd integers −(2n + 1) is defined by

Remark 2.1: 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

Main results and their proofs
Now we are in a position to state our main results and to prove them simply.
Proof: This proof is a slight modification of the first part in the second proof of [21, Theorem 1.1].
Taking f (u) = 2/(1 + u) and u = h(x) = √ 1 − 4x in the formula (6) and utilizing the identity (7) yield for n ∈ N. Further making use of the formula (8) and simplifying arrive at The proof of Theorem 3.1 is complete.

Remark 3.1:
Comparing (2) with (9) derives This expression is quite simpler, more easily remembered, more possibly understood, more readily computed, more significant, and more meaningful than the one in (3).

Theorem 3.2:
For n ∈ N, the power to n and the derivatives of the generating function G(x) defined in (1) satisfy Proof: The derivative formula (9) 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 (5)!!! Remark 3.3: This paper is a shortened version of the preprint [23].