Pattern 1^{j+1}0^j avoiding binary words

In this paper we study the enumeration and the construction of particular binary words avoiding the pattern 1^{j+1}0^j. By means of the theory of Riordan arrays, we solve the enumeration problem and we give a particular succession rule, called jumping and marked succession rule, which describes the growth of such words according to their number of ones. Moreover, the problem of associating a word to a path in the generating tree obtained by the succession rule is solved by introducing an algorithm which constructs all binary words and then kills those containing the forbidden pattern.