Pyth, 8
/.cyQQhQ
Try it online or run the Test Suite
Explanation
/.cyQQhQ ## implicit: Q = eval(input())
/ hQ ## integer division by (Q + 1)
.c ## nCr
yQ ## use Q * 2 as n
Q ## use Q as r
/.cyQQhQ
Try it online or run the Test Suite
/.cyQQhQ ## implicit: Q = eval(input())
/ hQ ## integer division by (Q + 1)
.c ## nCr
yQ ## use Q * 2 as n
Q ## use Q as r