1 Introduction

2 Combinatorial manifestations of (2n − 1)!!
2.1 Trapezoidal words
2.2 Perfect matchings
2.3 Stirling permutations
2.4 Increasing ordered trees
2.5 Leaf-labeled 0-2 trees
2.6 Height-labeled Dyck paths
2.7 Height-labeled ordered trees
2.8 Overhang paths

3 Pfaffians, Hafnians and Dyck paths

4 Round-Number Identities

5 Non-Round Identities

6 Refinements
6.1
6.2
6.3