Recommending API sequences is crucial in software development, saving developers time and effort. While previous studies primarily focus on accuracy, often recommending popular APIs, they tend to overlook less frequent, or ”tail,” APIs. This oversight, often a result of limited historical data, consequently diminishes the diversity of recommender systems. In this article, we propose DDASR, a framework for recommending API sequences containing both popular and tail APIs. To accurately capture developer intent, we utilize recent Large Language Models for learning query representations. To gain a better understanding of tail APIs, DDASR clusters tail APIs with similar functionality and replaces them with cluster centers to produce a pseudo ground truth. Moreover, a loss function is defined based on learning-to-rank to achieve an equilibrium in accuracy and diversity due to the inherent tradeoff between them. To evaluate DDASR, we conduct extensive experiments on Java and Python open source datasets. Results demonstrate that DDASR significantly achieves the best diversity without sacrificing accuracy. Compared to seven state-of-the-art approaches, DDASR improves accuracy metrics BLEU, ROUGE, MAP, and NDCG and diversity metric coverage by 108.28%, 67.30%, 88.59%, and 45.83%, respectively, on the Java dataset, as well as 9.83%, 2.45%, 8.06%, and 8.03%, respectively, on the Python dataset.