Count, Replace, Add Up!

MATL, 5 bytes

EGqXn

Try it online, or verify all test cases!

Explanation:

E       #Double the input
 Gq     #Push "input - 1"
   Xn   #Calculate "nchoosek" on the two numbers

Python, 65 bytes

def f(k):
 n,p=2*k,1
 for i in range(1,k):p=p*n//i;n-=1
 return p

Explanation: It's a simplified version of n choose k. See this sequence.