A critical limitation of conventional sequential recommendation models (SRMs) is their reliance on observed user-item interaction sequences within a closed-world setting, which hinders their ability to generalize to unseen or infrequent items. Recently, Large Language Models (LLMs) have shown remarkable promise in recommendation systems due to their vast world knowledge and advanced reasoning capabilities. Current research has predominantly explored two approaches: using LLMs to directly generate recommendations and distilling knowledge from LLMs to enhance conventional SRMs. However, these approaches face two major challenges: (1) high inference costs, as they require LLM responses during inference, either for generating predictions or as supplementary input; (2) inadequate distillation of the reasoning process, as existing methods focus mainly on improving embeddings or aligning outputs, without fully integrating LLMs' inherent reasoning capabilities. To address these issues, we propose LCKD-SR, anLLM-drivenCascadedKnowledgeDistillation framework forSequentialRecommendation. In this framework, an LLM, a Teacher SRM, and a Student SRM form a hierarchical distillation structure, enabling an LLM-free inference by using only the Student model. Beyond traditional embedding and ranking distillation, our framework abstracts the LLM's sequential reasoning abilities by identifying key interactions that subsequently guide the Teacher's attention using learnable markers. The Student model, which mirrors the architecture of the Teacher, achieves seamless knowledge alignment from the Teacher across all three aspects. Extensive experiments demonstrate the effectiveness and efficiency of the proposed LCKD-SR, showcasing its scalability to perform multi-level knowledge transfer while enabling LLM-independent inference, thereby overcoming the inference cost and reasoning limitations of existing methods.