Number of Different AVL Tree
Let $a_{n,h}$ denote the number of AVL trees with $n$ nodes and height $h$. It is straightforward to get a recurrence for $a_{n,h}$:

$$a_{n,h} = \sum_{k=1}^n \bigl(a_{k-1,h-1}a_{n-k,h-1} + a_{k-1,h-1}a_{n-k,h-2} + a_{k-1,h-2}a_{n-k,h-1}\bigr),~ n\geq h &gt; 1,$$

with the initial conditions $a_{n,h} = 0$, if $h&gt;n$ or $h\in\{0,1\}, n\neq h$, and $a_{0,0} = a_{1,1} = 1$.

The numbers you are looking for then are $a_n = \sum_h a_{n,h}$.

Unfortunately there apparently is no known closed form for this sequence. The OEIS has a list of the first 1000 terms and maple + mathematica code to compute further ones (via the recursion).