Number in the bracket sequence

A correct bracket sequence of length $2n$ is a sequence that can appear in some arithmetic expression. For example, ()()() and (())() are correct bracket sequences, while (((()) and (()))( are not.

All bracket sequences can be arranged in lexicographical order, where ( is considered less than ). For example, when n=3, the ordered list of correct bracket sequences is: ((())), (()()), (())(), ()(()), ()()().

Your task is to determine the lexicographical number (starting from zero) of a given correct bracket sequence of length $2n$.