Sequence A000085 in the On-Line Encyclopedia of Integer Sequences counts the number $A_n$ of involutions on $n$ letters, and also, the number of Young tableaux with $n$ cells.
I am curious, what is the limit of $A_n/A_{n-1}$ as $n\to\infty$ ?
How do you figure that out?
And, if it exists, is it a number of any special significance?
The limit does not exist. For example note the recurrence
$$A_n = A_{n-1} + (n-1) A_{n-2}$$
from which you can see that $A_n > (n-1) A_{n-2}$. However it appears that $$\lim_{n \to \infty} {A_n \over \sqrt{n} A_{n-1}} = 1.$$ This should be provable from the asymptotic formula (given in the OEIS entry)
$$ A_n \sim {1 \over \sqrt{2}} * \exp \left( \sqrt{n} - {n \over 2} - {1 \over 4} \right) n^{n/2}. $$