>How many Latin Squares of order-N are there?

From the On-Line Encyclopedia of Integer Sequences at
http://www.research.att.com/~njas/sequences/index.html,
for N=1 to 10, there are respectively (sequence A002860):

1,2,12,576,161280,812851200,61479419904000,108776032459082956800,
5524751496156892842531225600,9982437658213039871725064756920320000.

>  Is there a known explicit or recursive formula?

Apparently not, as EIS labels this sequence as "hard".