Time complexity to find out the number of ways to parenthesize N matrices
These are called Catalan numbers, and the term $\binom{2N}{N}$ is called a central binomial coefficient.

There are several formulas for these numbers which can be used to compute them efficiently, such as the one you present, or

$$C_n = \prod_{k=2}^n \frac{n+k}{k}$$

The problem with writing down the time complexity for computing these numbers is that the Catalan numbers grow exponentially&mdash;the number of bits in the output is roughly proportional to the input index. Therefore, the exact time complexity for evaluating these numbers from such a formula is going to depend significantly on the time complexity of your arithmetic operations, especially your choice of multiplication algorithm for large numbers.