On the r-Shifted Central Coefficients of Riordan Matrices

By presenting Riordan matrix as a triangle, the central coefficients are entries in the central column. Starting at the central column, the r-shifted central coefficients are entries in column r of the right part of the triangle.This paper aims to characterize the r-shifted central coefficients of Riordan matrices. Here we will concentrate on four elements of the subgroups of the Riordan group, that is, the Bell subgroup, the associated subgroup, the derivative subgroup, and the hitting time subgroup. Some examples are presented to show how we deduce the generating functions for interesting sequences by using different means of calculating these r-shifted central coefficients. Besides, we make some extensions in the Bell subgroup.


Introduction
Every infinite lower triangular array, in particular Riordan array, can be presented as a triangle.For instance, the Pascal triangle can be represented in the following form: ) . (2) We will refer to the former representation as the ISO (isosceles) triangle and to the latter as the REC (rectangular) triangle.The central coefficients are given by the central column of the ISO triangle, which play an important role in combinatorics.Barry [1] makes use of the central coefficients of Bell matrices to deduce the generating functions of interesting sequences.A natural question to ask is, "What happens when other columns of the ISO triangle are examined?"We investigate this question for the right part of the Riordan triangle.To consider this case we define the -shifted central coefficients and give their computing methods in four subgroups of the Riordan group in Section 2. With the -shifted central coefficients, we can deduce the generating functions of more sequences.Also, some examples are presented to show how we deduce the generating functions of interesting sequences by using different means of calculating these shifted central coefficients.Some sequences referred to by their Annnnnn OEIS number can be found in On-Line Encyclopedia of Integer Sequence (OEIS) [2].Furthermore, we introduce the AER (aerated) triangle as the ISO form with interposed zeros: 1 1 0 1 1 0 2 0 1 1 0 3 0 3 0 1 1 0 4 0 6 0 4 0 1 1 0 5 0 10 0 10 0 5 0 1 1 0 6 0 15 0 20 0 15 0 6 0 1 ⋅ ⋅ ⋅ 2

Journal of Applied Mathematics
Considering a Riordan array in its AER form, we prove that the right part of the AER triangle is an aerated Riordan array.That is to say, the -shifted central coefficients with interposed zeros can generate an aerated Riordan array and the relation of this Riordan array with the initial one will be shown in Section 3. Finally, we defined the (, )-shifted central coefficients for expanding our research subject.Before defining the -shifted central coefficients we recall the Riordan group [3], sate the Fundamental Theorem of Riordan arrays, and give four of the subgroups of the Riordan group.Consider infinite matrices  = ( , ) ,≥0 with entries in C, the complex numbers.Let   () = ∑ ∞ ≥0  ,   be the generating function of the th column of .We now make the crucial special assumption that where In this case, we write  = ((), ℎ()) and say that  is a Riordan array.That is to say, the pair ((), ℎ()) defines the  = ( , ) ,≥0 having Suppose we multiply  = ((), ℎ()) by a column vector ( 0 ,  1 , . . . )  and the result is the column vector ( 0 ,  1 , . . . )  .If the generating function for the sequence ( 0 ,  1 , . . . )  is () and similarly, ( 0 ,  1 , . . . )  has () as its generating function, then we obtain  () =  ()  (ℎ ()) .(7) This is called the Fundamental Theorem of Riordan arrays.The typical column of  = ((), ()) is ()[()]  and using this as () quickly yields the matrix multiplication rule for the Riordan group which is This shows us that the identity  = (1, ), the usual matrix identity, and group inverse where ℎ() is the compositional inverse of ℎ(), such that ℎ(ℎ()) = ℎ(ℎ()) = .In addition, subgroups of the Riordan group are important and have been considered in the literature.
In order to compute the -shifted central coefficients of matrices in the previous defined subgroups, we need the Lagrange Inversion Formula, whose proof can be found in [7].
Lemma 1 (LIF).Let ℎ() be a formal power series with ℎ(0) = 0 and ℎ  (0) ̸ = 0 and let ℎ() be its compositional inverse; then one has According to this definition, we find that the central coefficients are equal to 0-shifted central coefficients.Sometimes, we ignore the central coefficients; that is to say, the first column considered is 1-shifted central coefficients.Therefore, ( + 1)-shifted central coefficients should be given for  ∈ N.
In this section, we characterize the -shifted central coefficients of the matrices in four subgroups, of which the Bell subgroup is the most important.Also, some examples are presented to show how we deduce the generating functions of interesting sequences by using different methods to calculate these -shifted central coefficients.
Proof.According to formula (6), we can calculate  2+,+ directly, We now apply the LIF.This says that if we have then Thus we obtain that where () =  2 /ℎ(); then formula (12) is obtained.
From formula (12) and formula (13) can be obtained immediately.
Here we just list the first two cases: in the case  = 0, the sequence we give is little Schröder numbers A001003, and in the case  = 1, the sequence we give begins 1, 3, 14, 70, 363, . . .
Making use of -sequence and Theorem 4, we obtain a new characterization to the generating functions for central coefficients of a Bell-type array.Now we show this result as a corollary.
Then the result follows by using Theorem 4.
Proof.First of all, we have According to the Lagrange Inversion Formula, we have since we set Therefore, we get where () =  2 /ℎ().
We wish to show that where  = √ 1 − 6 +  2 ,  ∈ N. We can call this ( + 1)-fold convolution of the large Schroeder numbers.We list the first few cases: when  = 0, the sequence is large Schröder numbers A006318, when  = 1, the sequence is A006319, and when  = 2, the sequence is A006320.

(𝑚,𝑟)-Shifted Central Coefficients.
For expanding our research subject in Bell subgroup, we now repeat the following steps.
(i) Stretch the infinite lower triangular array so that it becomes isosceles.
(ii) Consider the columns of the right part of the ISO triangle.
(iii) Regard the right part of the ISO triangle as an infinite lower triangular array.
We can repeat this process infinite times, because the right part of every triangle can be regarded as an infinite lower triangular array.Considering a Bell-type array  as the initial one, we now repeat this process  = 1, 2, 3, . . .times for the initial array; then we should consider the (, )-shifted central coefficients defined as the sequence  (+1)+,+ , where ,  ∈ N,  = 1, 2, 3, . .., just like the following cases.

Theorem 4 .
Let  = (ℎ()/, ℎ()) be an element of the Bell subgroup of the Riordan group.If  2+,+ denote the -shifted central coefficients of , then one has