Related papers: Computationally Efficient Bounds for the Sum of Ca…
We provide a context around a conjectured closed form for the Hankel transform of linear combinations of consecutive pairs of Catalan numbers. This generalizes the formula for the Hankel transforms of the shifted Catalan numbers and the…
In this paper we establish some new congruences involving central binomial coefficients as well as Catalan numbers. Let $p$ be a prime and let $a$ be any positive integer. We determine $\sum_{k=0}^{p^a-1}\binom{2k}{k+d}$ mod $p^2$ for…
It is well known that the Catalan number C_n counts dissections of a regular (n+2)-gon into triangles. Here we count such dissections by number of triangles that contain two sides of the polygon among their three edges, leading to a…
We construct a new tail bound for the sum of independent random variables for situations in which the expected value of the sum is known and each random variable lies within a specified interval, which may be different for each variable.…
We discuss the properties of the Hankel transformation of a sequence whose elements are the sums of consecutive generalized Catalan numbers and find their values in the closed form.
The masses of octet baryons are calculated by the method of QCD sum rules. Using generalized interpolating fields, three independent sets of QCD sum rules are derived which allow the extraction of low-lying N* states with spin-parity 1/2+,…
A new class of alternating convolutions concerning binomial coefficients and Catalan numbers are evaluated in closed forms.
The Catalan numbers are well-known to be the answer to many different counting problems, and so there are many different families of sets whose cardinalities are the Catalan numbers. We show how such a family can be given the structure of a…
The Catalan transform of a sequence (a_{n})_{n>=0} is the sequence (b_{n})_{n>=0} with b_{n} = Sum[k/(2n-k) (2n-k)-choose-(n-k) a_{k},k=0..n]. Here we show that the Catalan transform of the Catalan numbers has a simple interpretation: it…
This paper presents a lower bound for optimizing a finite sum of $n$ functions, where each function is $L$-smooth and the sum is $\mu$-strongly convex. We show that no algorithm can reach an error $\epsilon$ in minimizing all functions from…
We obtain new lower and upper bounds for probabilities of unions of events.These bounds are sharp. They are stronger than earlier ones. General bounds maybe applied in arbitrary measurable spaces.We have improved the method that has been…
We present a $q$-analog of the super Catalan number $(2m)!(2n)!/2m!n!(m+n)!$, which also generalizes the $q$-Catalan numbers $c_n(\lambda)$, due to F\"urlinger and Hofbauer, for $\lambda=0$ and $\lambda=1$. We give a combinatorial…
A new general and unified method of summation, which is both regular and consistent, is invented. It is based on the idea concerning a way of integers reordering. The resulting theory includes a number of explicit and closed form summation…
We give explicit bounds for the tail probabilities for sums of independent geometric or exponential variables, possibly with different parameters.
Lower and upper bounds are explored for the uniform (Kolmogorov) and $L^2$-distances between the distributions of weighted sums of dependent summands and the normal law. The results are illustrated for several classes of random variables…
Recent results about sums of cubes of Fibonacci numbers [Frontczak, 2018] are extended to arbitrary powers.
The goal of this work is to prove a new sure upper bound in a setting that can be thought of as a simplified function field analogue. This result is comparable to a recent result of the author concerning almost sure upper bound of random…
New bounds on the cardinality of permutation codes equipped with the Ulam distance are presented. First, an integer-programming upper bound is derived, which improves on the Singleton-type upper bound in the literature for some lengths.…
An asymptotic formula for the sum of the first n primes is derived. This result improves the previous results of P. Dusart. Using this asymptotic expansion, we prove the conjectures of R. Mandl and G. Robin on the upper and the lower bound…
In this note we introduce a determinant and then give its evaluating formula. The determinant turns out to be a generalization of the well-known ballot and Fuss-Catalan numbers, which is believed to be new. The evaluating formula is proved…