On a way of division of segments (Q1096094)

The authors study the recurrent equation (1) \(a_{n,k}=a_{n+1,k}+a_{n+1,k+1}\), \(k=1,2,...,n\), \(n\in N\) with the initial condition \(a_{3,1}=u\), \(a_{3,2}=v\), \(a_{3,3}=w\). They prove that if u, v, w are positive numbers, then there exists a sequence \(a_{n,k}\) of positive numbers satisfying equation (1) if and only if \(v^ 2\leq uw\). The sequence is unique if and only if \(v^ 2=uw\). The authors show that the theorem above can be deduced from some results of \textit{M. Laczkovich}, obtained by using advanced tools [Non-negative measurable solutions of difference equations, J. London Math. Soc. 2, 139-147 (1986)] while the proof they present is completely elementary.