Relations

How many of the nn2 binary operations are associative?
 (x⊙y)⊙z=x⊙(y⊙z)

https://mathoverflow.net/questions/110211/how-many-binary-operations-are-associative

for n=2, exactly half of the 24=16 binary operations are associative.
But for n=3, only 113 of the 39=19,683 binary operations are associative, a count I do not trust, because it seems so much smaller than I anticipated.
It is difficult to count among the four billion (4,294,967,296) binary operations for n=4.) the OEIS sequence A023814. The n=4 number I couldn't easily compute is 3492.

I would be interested in the asymptotic growth rate. Surely this is all well known...? Thanks for pointers! 
Update. Following MSE link provided by Darij, I reached (via Gerry Myerson's pointer) 

http://oeis.org/search?q=1%2C2%2C3%2C6%2C11%2C23%2C47%2C106%2C235&language=english&go=Search

Is the Ratio of Associative Binary Operations to All Binary Operations on a Set of n Elements Generally Small?