Abstract
In this chapter, we consider the Catalan numbers, C n = 1 n + 1 2 n n , and two of their generalizations, Catalan triangle numbers, B n , k and A n , k , for n , k ∈ N . They are combinatorial numbers and present interesting properties as recursive formulae, generating functions and combinatorial interpretations. We treat the moments of these Catalan triangle numbers, i.e., with the following sums: ∑ k = 1 n k m B n , k j , ∑ k = 1 n + 1 2 k − 1 m A n , k j , for j , n ∈ N and m ∈ N ∪ 0 . We present their closed expressions for some values of m and j . Alternating sums are also considered for particular powers. Other famous integer sequences are studied in Section 3, and its connection with Catalan triangle numbers are given in Section 4. Finally we conjecture some properties of divisibility of moments and alternating sums of powers in the last section.
Highlights
After the binomial coefficients, the well-known Catalan numbers ðCnÞn ≥ 0 are the most frequently occurring combinatorial numbers
Other applications of the Catalan numbers appear in engineering in the field of cryptography to form keys for secure transfer of information; in computational geometry, they are generally used in geometric modeling; they may be found in geographic information systems, geodesy, or medicine
The proof which we present here allows to recognize the natural connection among the sequences ðaðnÞÞn ≥ 0 and ðbðnÞÞn ≥ 1 and the Catalan numbers ðCnÞn ≥ 0
Summary
The well-known Catalan numbers ðCnÞn ≥ 0 are the most frequently occurring combinatorial numbers. They are treated deeply in many books, monographs, and papers (e.g., [1–20]). Catalan numbers play an important role and have a major importance in computer science and combinatorics They appear in studying astonishingly many combinatorial problems. In ([9], Theorem 1.1), the authors show that any binomial coefficient can be written as weighted sums along the rows of the Catalan triangle, i.e.,. We present alternating sums of for several powers of Catalan triangle numbers (Theorem 2.2, Proposition 4.1 (iii), and Proposition 4.4 (iii)). In Conjecture 5.3, we state that the factor nþ
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