Recurrent neural networks empirically generate natural language with high syntactic fidelity. However, their success is not well-understood theoretically. We provide theoretical insight into this success, proving in a finite-precision setting that RNNs can efficiently generate bounded hierarchical languages that reflect the scaffolding of natural language syntax. We introduce Dyck-(k,m), the language of well-nested brackets (of k types) and m-bounded nesting depth, reflecting the bounded memory needs and long-distance dependencies of natural language syntax. The best known results use O(k^m⁄₂) memory (hidden units) to generate these languages. We prove that an RNN with O(m log k) hidden units suffices, an exponential reduction in memory, by an explicit construction. Finally, we show that no algorithm, even with unbounded computation, can suffice with o(m log k) hidden units.