Moments of combinatorial and Catalan numbers

Miana, Pedro J.; Romero, Natalia

doi:10.1016/j.jnt.2010.01.018

Cited by 21 publications

(17 citation statements)

References 8 publications

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“…The numbers in the first column of the admissible matrix are called Catalan-like numbers, which are investigated in [5] from combinatorial views. The admissible matrix A = (A n,k ) n k 0 associated to the Catalan triangle B is defined by A n,k = 2k+1 2n+1 2n+1 n−k , which is considered by Miana and Romero [38] by evaluating the moments Φ m = n k=0 (2k + 1) m A 2 n,k . Table 1.2 illustrates this triangle for small n and k up to 6.…”

Section: Introductionmentioning

confidence: 99%

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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%

“…The triangle C is also called the "Catalan triangle" in the literature, despite it has the most-standing form C = (C n,n−k ) n k 0 first discovered in 1961 by Forder [24], see for examples [1,6,9,23,30,38,47]. Table 1.3 illustrates this triangle for small n and k up to 7.…”

Section: Introductionmentioning

confidence: 99%

Some New Binomial Sums Related to the Catalan Triangle

Sun

2014

Electron. J. Combin.

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“…These sums satisfy a much simpler recurrence relation than the coefficients b l a,k themselves, where B a,k = B −a,k as a result of the symmetry property (4.3). [20,21].…”

Section: Moment Analysis On the Diagonalmentioning

confidence: 99%

A semicircle law and decorrelation phenomena for iterated Kolmogorov loops

Habermann

2020

J. London Math. Soc.

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We consider a standard one-dimensional Brownian motion on the time interval [0,1] conditioned to have vanishing iterated time integrals up to order N. We show that the resulting processes can be expressed explicitly in terms of shifted Legendre polynomials and the original Brownian motion, and we use these representations to prove that the processes converge weakly as N → ∞ to the zero process. This gives rise to a polynomial decomposition for Brownian motion. We further study the fluctuation processes obtained through scaling by √ N and show that they converge in finite-dimensional distributions as N → ∞ to a collection of independent zero-mean Gaussian random variables whose variances follow a scaled semicircle. The fluctuation result is a consequence of a limit theorem for Legendre polynomials which quantifies their completeness and orthogonality property. In the proof of the latter, we encounter a Catalan triangle.

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“…appear as the entries of this second Catalan triangle, which is considered in [11]. Notice that A n,1 = C n and C 2n+1,n−k+1 = A n,k for k ≤ n + 1.…”

Section: Recurrence Relation and Sums Of Catalan Triangle Numbersmentioning

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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Moments of combinatorial and Catalan numbers

Cited by 21 publications

References 8 publications

Some New Binomial Sums Related to the Catalan Triangle

Some New Binomial Sums Related to the Catalan Triangle

A semicircle law and decorrelation phenomena for iterated Kolmogorov loops

Sums of powers of Catalan triangle numbers

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