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By integrating constructs from the lambda-calculus and the pi-calculus, in higher order process calculi exchanged values may contain processes. This paper studies the relative expressiveness of HO pi, the higher-order pi-calculus in which communications are governed by session types. Our main discovery is that HO, a subcalculus of HO pi which lacks name-passing and recursion, can serve as a new core calculus for session-typed higher-order concurrency. We show that HO can encode HO pi fully abstractly (up to typed contextual equivalence) more precisely and efficiently than the first-order session pi-calculus (pi). Overall, under the discipline of session types, HO pi, HO, and pi are equally expressive; however, we show that HO pi is more tightly related to HO than to pi. (C) 2019 Elsevier Inc. All rights reserved.