I have been playing around with interesting integer sequences and came across [Schröder number][1] which defines the number of lattice paths of n x n grid.

The recurrence formula to calculate these numbers is as follows:

$$
S_n = 3S_{n-1} + \sum_{k=1}^{n-2}S_kS_{n-k-1} \text{  for  }n\ge2 \text{ with } S_0 =1, S_2=2 
$$
The first few numbers in the sequence are:
$$
1, 2, 6, 22, 90, 394, 1806, 8558
$$
I was curious about the following ratio and its limit:
$$
\lim_{n\rightarrow\infty}\frac{S_{n-1}}{S_{n}} = ? 
$$
So I wrote a small C++ program with mixed-precision to handle the large numbers and surprisingly there appears to be an asymptote that is not zero. So far my little program has reached n = 20,000, but it's slowing down significantly as the numbers are getting truly large but it has given a preliminary result of
$$
\lim_{n\rightarrow\infty}\frac{S_{n-1}}{S_{n}} = 0.17158... 
$$

My question now is, is this simply a ghost limit for not having gone far enough, or does a limit exist that is not zero? My math expertise is lacking somewhat to attempt to find that limit analytically (can it even be done?).

If a limit exists would it imply that, for large n, the number of additional lattice paths of a n x n grid only increases by about 17% by expanding the grid by 1?




  [1]: https://en.wikipedia.org/wiki/Schr%C3%B6der_number