Post Number:#2 by Phlembac Adib Hasan » Sat Jan 03, 2015 11:33 am
Start the sequence from $a_0$ and toss out the first term. So the third condition turns into $a_i\leq i-1$. Now consider the following bijection: take any $A$ dominated sequence of length $n$ and set $a_i$ as the number of $B$'s preceding $i$-th $A$. Since the sequence is $A$ dominating, there will be at most $i-1$ $B$'s before $i$-th $A$. And obviously the derived sequence will be non-decreasing.
Also, from a given sequence of $a_i$s, it is possible to reverse this move and make an $A$ dominated sequence. Thus the number of sequences is $C_n=\frac{1}{n+1}\binom {2n}{n}$
And here are two examples of the bijection, for clarification:
$AABABB\to 0,0,1$
$AABBAB\to 0,0,2$