Many sequences of binomial coefficients share various unimodality properties. In this paper we consider the unimodality problem of a sequence of binomial coefficients located in a ray or a transversal of the Pascal triangle. Our results give in particular an affirmative answer to a conjecture of Belbachir et al which asserts that such a sequence of binomial coefficients must be unimodal. We als...

for any n ≥ 0 and 0 ≤ k ≤ n. A combinatorial interpretation of the formula (1) is as follows. The left-hand side counts the number of ways to select k balls (numbered with the corresponding number) from a bin of n balls. This is equivalent to the sum of the following two cases: (a) if the ball n is selected, the number of ways to select the remaining k − 1 balls from n − 1, and (b) if the ball ...

This paper, using computational and theoretical methods, deals with prime divisors of binomial coefficients: Geometric distribution and number of distinct prime divisors are studied. We give a numerical result on a conjecture by Erdôs on square divisors of binomial coefficients.

We discuss divisibility properties of some differences of the central binomial coefficients and Catalan numbers. The main tool is the application of various congruences modulo high prime powers for binomial coefficients combined with some recurrence relevant to these combinatorial quantities.

In this paper, we compute certain sums involving the inverses of binomial coefficients. We derive the recurrence formulas for certain infinite sums related to the inverses of binomial coefficients.

In this paper we obtain a universal lower bound on the 2-adic order of lacunary sums of binomial coefficients. By means of necessary and sufficient conditions, we determine the set of values for which the bound is achieved and show the periodicity of the set. We prove a congruential identity for the corresponding generating function. Our approach gives an alternative and transparent proof for s...