The Latin Square Count Challenge

When I started to learn about latin squares, there was this question about them to which I have not found an answer yet:

How many Latin Squares are there of size n ?

I am not asking for a tabulated listing of different sizes & the count of latin squares for each of those sizes. With fast computers available today, I guess, programmers & others have used brute force approach to get the count (by trying out combinations one after another programatically) for considerably large sizes. Instead, what I am looking for is an elegant, closed, algebraic formula (involving n, the size).

I dont know if it is a tough job or whether no one is interested in solving it. Atleast, I have not found a solution yet. Besides, one may ask what's the use of it after all.

Forget the count for now, can someone atleast prove the following for me...

There are more latin squares of size n+1 than those of size n

Though it looks obvious, a rigid mathematical proof eludes me.