Generator is a concise representation for frequent itemset. And it has the anti-monotone property as the frequent itemset does, which is an important property in real applications. But when itemsets are attached with weights to balance importances between themselves, the anti-monotone property of generator may not hold. Additionally generator with weight may become tough to be dealt with in many circumstances. In this paper, we adapt support weight calculation to generator definition under weight support framework through specific techniques. The anti-monotone property of generator with weight can be kept to facilitate mining works. A new method for mining generators with weights is proposed. It exploits depth-first mining strategy and prunes search space with little cost. Experimental results show that the proposed method runs properly and achieves good performance.