Mining frequent sequential patterns consists in extracting recurrent behaviors, modeled as patterns, in a big sequence dataset. Such patterns inform about which events are frequently observed in sequences, i.e. what does really happen. Sometimes, knowing that some specific event does not happen is more informative than extracting a lot of observed events. Negative sequential patterns (NSP) formulate recurrent behaviors by patterns containing both observed events and absent events. Few approaches have been proposed to mine such NSPs. In addition, the syntax and semantics of NSPs differ in the different methods which makes it difficult to compare them. This article provides a unified framework for the formulation of the syntax and the semantics of NSPs. Then, we introduce a new algorithm, NegPSpan, that extracts NSPs using a PrefixSpan depth-first scheme and enabling maxgap constraints that other approaches do not take into account. The formal framework allows for highlighting the differences between the proposed approach wrt to the methods from the literature, especially wrt the state of the art approach eNSP. Intensive experiments on synthetic and real datasets show that NegPSpan can extract meaningful NSPs and that it can process bigger datasets than eNSP thanks to significantly lower memory requirements and better computation times.