Catalan matrix and related combinatorial identities

Stanimirović, Stefan; Stanimirović, Predrag S.; Miladinović, Marko; Ilić, Aleksandar

doi:10.1016/j.amc.2009.06.003

Cited by 11 publications

(5 citation statements)

References 8 publications

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“…We used the notion of the lower triangular diagonally constant Catalan matrix C n [x] from [14], whose nonzero elements are Catalan numbers. Motivated by the main result in [1], where it is shown how to invert the linear combination I − λP n [x] of the Pascal matrix with the identity matrix, in the present paper we invert various linear combinations of the Catalan matrix with the identity matrix.…”

Section: Resultsmentioning

confidence: 99%

Inversion of Catalan matrix plus one

Stanimirović

2010

J. Appl. Math. Comput.

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Motivated by a statistical problem, in (Aggarwala and Lamoureux in Am. Math. Mon., 109:371-377, 2002) it is shown how to invert a linear combination of the Pascal matrix with the identity matrix. Continuing this idea, we invert various linear combinations of the Catalan matrix, introduced in (Stanimirović et al. in Appl. Math. Comput. 215:796-805, 2009), with the identity matrix. In (Aggarwala and Lamoureux in Am. Math. Mon., 109:371-377, 2002) the occurrence of the polylogarithm function is observed. Inverses of linear combinations of the Catalan and the identity matrix are expressed in terms of Catalan numbers, the pochhammer function and the generalized hypergeometric function.

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Section: Resultsmentioning

confidence: 99%

Inversion of Catalan matrix plus one

Stanimirović

2010

J. Appl. Math. Comput.

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“…Following this idea, some combinatorial identities via Fibonacci numbers were derived in [4,14], as well as the identities for the Catalan numbers [9], Bell [10], Bernoulli [13] and the Lucas numbers [15] were also computed. In [5] the authors derived identities by using the factorizations of the Pascal matrix via generalized second order recurrent matrix.…”

Section: Introductionmentioning

confidence: 99%

A matrix approach to the binomial theorem

Stanimirović

2013

Ukr Math J

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Motivated by the formula x n = n k=0 n k (x − 1) k , we investigate factorizations of the lower triangular Toeplitz matrix with (i, j)th entry equal to x i−j via the Pascal matrix. In this way, a new computational approach to a generalization of the binomial theorem is introduced. Numerous combinatorial identities are obtained from these matrix relations. На основi формули x n = n k=0 n k (x − 1) k розглянуто факторизацiї нижньотрикутної матрицi Теплiца, (i, j)й елемент якої дорiвнює x i−j , з використанням матрицi Паскаля. Тим самим уведено новий обчислювальний пiдхiд до узагальнення бiномiальної теореми. Iз використанням цих матричних спiввiдношень отримано численнi комбiнаторнi тотожностi.

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“…The Catalan numbers arise in many combinatorial problems -see Stanley [21] for several combinatorial interpretations of these numbers, and see [20] for matrix approach. These numbers also satisfy the recurrence relation C 0 = 1 and…”

Section: Introductionmentioning

confidence: 99%

On the number of restricted Dyck paths

Ilić

Ilic

2011

Filomat

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In this note we examine the number of integer lattice paths consisting of up-steps (1, 1) and down-steps (1, −1) that do not touch the lines y = m and y = −k, and in particular Theorem 3.2 in [P. Mladenović, Combinatorics, Mathematical Society of Serbia, Belgrade, 2001]. The theorem is shown to be incorrect for n m + k + min(m, k), and using similar combinatorial technique we proved the upper and lower bound for the number of such restricted Dyck paths. In conclusion, we present some relations between the Chebyshev polynomials of the second kind and generating function for the number of restricted Dyck paths, and connections with the spectral moments of graphs and the Estrada index.

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Catalan matrix and related combinatorial identities

Cited by 11 publications

References 8 publications

Inversion of Catalan matrix plus one

Inversion of Catalan matrix plus one

A matrix approach to the binomial theorem

On the number of restricted Dyck paths

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