Several identities in the Catalan triangle

Zhang, Zhizheng; Bi-jun, Pang

doi:10.1007/s13226-010-0022-0

Cited by 9 publications

(8 citation statements)

References 10 publications

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“…Some alternating sum identities on the Catalan triangle B were established by Zhang and Pang [53], who showed that the Catalan triangle B can be factorized as the product of the Fibonacci matrix and a lower triangular matrix, which makes them build close connections among C n , B n,k and the Fibonacci numbers. Motivated by a matrix identity related to the Catalan triangle B [46], Chen et al [14] derived many nice matrix identities on weighted partial Motzkin paths.…”

Section: Introductionmentioning

confidence: 99%

Some New Binomial Sums Related to the Catalan Triangle

Sun

2014

Electron. J. Combin.

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In this paper, we derive many new identities on the classical Catalan triangle $\mathcal{C}=(C_{n,k})_{n\geq k\geq 0}$, where $C_{n,k}=\frac{k+1}{n+1}\binom{2n-k}{n}$ are the well-known ballot numbers. The first three types are based on the determinant and the fourth is relied on the permanent of a square matrix. It not only produces many known and new identities involving Catalan numbers, but also provides a new viewpoint on combinatorial triangles.

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Section: Introductionmentioning

confidence: 99%

Some New Binomial Sums Related to the Catalan Triangle

Sun

2014

Electron. J. Combin.

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“…for j ¼ 2, 3 and m ∈ . For m ¼ 0, these identities are shown in [14,24]. See a unified proof in ([22], Corollary 2.2).…”

Section: Moments Of Squares and Cubes Of Catalan Triangle Numbersmentioning

confidence: 95%

“…are a new interesting research which could be considered in later articles, compared with ( [24], Theorem 2.3).…”

Section: Conclusion and Future Developmentsmentioning

confidence: 98%

Moments of Catalan Triangle Numbers

Miana¹,

Romero²

2020

Number Theory and Its Applications

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In this chapter, we consider the Catalan numbers, C n ¼ 1 nþ1 2n n , and two of their generalizations, Catalan triangle numbers, B n,k and A n,k , for n, k ∈ . 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: P n k¼1 k m B j n,k , P nþ1 k¼1 2k À 1 ðÞ m A j n,k , for j, n ∈  and m ∈  ∪ 0 fg. 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.

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“…Some of these equalities solve some open problems and connect Catalan sequences with other some known sequences, see for example Theorem 4.3. Note that we have not considered moments on these sums of powers as in other papers in the literature, see for example [3,11,23]. We have also presented a natural connection between harmonic numbers and Catalan triangle numbers which seems to be new and may be completed in later studies.…”

Section: New Conjectures Final Comments and Conclusionmentioning

confidence: 99%

Sums of powers of Catalan triangle numbers

Miana

Ohtsuka

Romero

2017

Discrete Mathematics

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In this paper we consider combinatorial numbers C m,k for m ≥ 1 and k ≥ 0 which unifies the entries of the Catalan triangles B n,k and A n,k for appropriate values of parameters m and k, i.e., B n,k = C 2n,n−k and A n,k = C 2n+1,n+1−k . In fact, some of these numbers are the well-known Catalan numbers Cn that is C2n,n−1 = C2n+1,n = Cn.We present new identities for recurrence relations, linear sums and alternating sum of C m,k . After that, we check sums (and alternating sums) of squares and cubes of C m,k and, consequently, for B n,k and A n,k . In particular, one of these equalities solves an open problem posed in [8]. We also present some linear identities involving harmonic numbers Hn and Catalan triangles numbers C m,k . Finally, in the last section new open problems and identities involving Cn are conjectured.

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Several identities in the Catalan triangle

Cited by 9 publications

References 10 publications

Some New Binomial Sums Related to the Catalan Triangle

Some New Binomial Sums Related to the Catalan Triangle

Moments of Catalan Triangle Numbers

Sums of powers of Catalan triangle numbers

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