Recent advances in high-throughput experimental technologies have generated a huge amount of data on interactions between proteins and nucleic acids. Motivated by the big experimental data, several computational methods have been developed either to predict binding sites in a sequence or to determine if an interaction exists between protein and nucleic acid sequences. However, most of the methods cannot be used to discover new nucleic acid sequences that bind to a target protein because they are classifiers rather than generators. In this paper we propose a generative model for constructing protein-binding RNA sequences and motifs using a long short-term memory (LSTM) neural network. Testing the model for several target proteins showed that RNA sequences generated by the model have high binding affinity and specificity for their target proteins and that the protein-binding motifs derived from the generated RNA sequences are comparable to the motifs from experimentally validated protein-binding RNA sequences. The results are promising and we believe this approach will help design more efficient in vitro or in vivo experiments by suggesting potential RNA aptamers for a target protein.