Mining frequent sequential patterns consists in extracting recurrent behaviors, modeled as subsequences, in a big sequence dataset. Such patterns inform about which events are frequently observed in sequences, i.e. events that really happen. Sometimes, knowing that some specific event does not happen is more informative than extracting observed events. Negative sequential patterns (NSPs) capture recurrent behaviors by patterns having the form of sequences mentioning both observed events and the absence of 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, N PS , that extracts NSPs using a PrefixSpan depth-first scheme, enabling maxgap constraints that other approaches do not take into account. The formal framework highlights the differences between the proposed approach and the methods from the literature, especially with the state of the art approach eNSP. Intensive experiments on synthetic and real datasets show that N PS can extract meaningful NSPs and that it can process bigger datasets than eNSP thanks to significantly lower memory requirements and better computation times.