We find a graded Artinian level O-sequence of the form $H;:;h_0;h_1;cdots;h_{d-1};h_dcdots$ $^{(d+1-1_)-st}h_d$ < $h_{d+s}$ not having the Weak-Lefschetz property. We also introduce several algorithms for construction of some examples of non-unimodal level O-sequences using a computer program called CoCoA.