Enumeration of the generalized Catalan numbers

The sequence of numbers given by the equation Cn=1n +12nn are widely known as the Catalan numbers, because they were first studied by Catalan. The ways to enumerate these numbers are similarly widely known and studied, as can be seen in Gould's catalog of Catalan numbers. Our objective is to find ways to enumerate this larger and generalized set of sequences, using multiple methods that will build on known enumerations of the Catalan numbers.