On a queer binomial sum

In this paper the binomial sum Sn(r) = n X m=0 µ n m ¶ (¡1) m m r + 1 (r > 0; n 2N) is investigated. It turns out that the behaviour of this sum for n ! 1 depends on the parameter r and changes dramatically at the values r = 1 and r = 2. In particular, for r ‚ 2 we obtain an oscillatory behaviour while for r < 2 the sequence Sn(r) is monotonous at least for large values of n.