On the number of outer connected dominating sets of graphs

Let $G=(V,E)$ be a simple graph. A set $S\subseteq V(G)$ is called an outer-connected dominating set (or ocd-set) of $G$, if $S$ is a dominating set of $G$ and either $S=V(G)$ or $V\backslash S$ is a connected graph. In this paper we introduce a polynomial which its coefficients are the number of ocd-sets of $G$. We obtain some properties of this polynomial and its coefficients. Also we compute this polynomial for some specific graphs.Comment: 1 pag

Variants of Schroeder Dissections

Some formulae are given for the enumeration of certain types of dissections of the convex (n+2)-gon by non-crossing diagonals. The classical Schroeder and Motzkin numbers are addressed using a cataloguing tool, the "reversive symbol". The elementary details are referred to three Web addresses.Comment: 2 page

On the denominators of harmonic numbers

Let $H_n$ be the $n$-th harmonic number and let $v_n$ be its denominator. It is well known that $v_n$ is even for every integer $n\ge 2$. In this paper, we study the properties of $v_n$. One of our results is: the set of positive integers $n$ such that $v_n$ is divisible by the least common multiple of $1, 2, \cdots, \lfloor {n^{1/4}}\rfloor$ has density one. In particular, for any positive integer $m$, the set of positive integers $n$ such that $v_n$ is divisible by $m$ has density one.Comment: 6 page

On sequences without geometric progressions

An improved upper bound is obtained for the density of sequences of positive integers that contain no k-term geometric progression.Comment: 4 pages; minor correctio

Sparse binary cyclotomic polynomials

We derive a lower and an upper bound for the number of binary cyclotomic polynomials $\Phi_m$ with at most $m^{1/2+\epsilon}$ nonzero terms.Comment: 3 page

Number Sequences in an Integral Form with a Generalized Convolution Property and Somos-4 Hankel Determinants

MSC 2010: 11B83, 05A19, 33C45This paper is dealing with the Hankel determinants of the special number sequences given in an integral form. We show that these sequences satisfy a generalized convolution property and the Hankel determinants have the generalized Somos-4 property. Here, we recognize well known number sequences such as: the Fibonacci, Catalan, Motzkin and SchrÄoder sequences, like special cases