In this paper we study the factors of some alternating sums of products of binomial and q-binomial coefficients. We prove that for all positive integers

The Ramanujan polynomials were introduced by Ramanujan in his study of power series inversions. These polynomials have been closely related to the enumeration of trees. In an approach to the Cayleyâ€¦ (More)

Let S m,n (q) := n k=1 1 âˆ’ q 2k 1 âˆ’ q 2 1 âˆ’ q k 1 âˆ’ q mâˆ’1 q m+1 2 (nâˆ’k). Generalizing the formulas of Warnaar and Schlosser, we prove that there exist poly-nomials P m,k (q) âˆˆ Z[q] such that S 2m+1,nâ€¦ (More)

By using the Newton interpolation formula, we generalize the recent identities on the Catalan triangle obtained by Miana and Romero as well as those of Chen and Chu. We further study divisibilityâ€¦ (More)

Two new expansions for partial sums of Gaussâ€™ triangular and square numbers series are given. As a consequence, we derive a family of inequalities for the overpartition function p(n) and for theâ€¦ (More)

Let In,k (respectively, Jn,k) be the number of involutions (respectively, fixed-point free involutions) of {1, . . . , n} with k descents. Motivated by Brentiâ€™s conjecture which states that theâ€¦ (More)

Generalizing a sequence of Lambert, Cayley and Ramanujan, Chapoton has recently introduced a polynomial sequence Qn := Qn(x, y, z, t) defined by Q1 = 1, Qn+1 = [x + nz + (y + t)(n + yâˆ‚y)]Qn. In thisâ€¦ (More)