#Japt, 16 bytes
Even Mathematica is shorter. :-/
U*2ª1 o àU l /°U
###Ungolfed and explanation
U*2ª 1 o àU l /° U
U*2||1 o àU l /++U
// Implicit: U = input number
U*2||1 // Take U*2. If it is zero, take 1.
o àU // Generate a range of this length, and calculate all combinations of length U.
l /++U // Take the length of the result and divide by (U+1).
// Implicit: output result
Alternate version, based on the recursive formula:
C=_?(4+6/~Z *C$(Z-1):1};$C(U