We know that equation $$s_1+s_2+s_3=n-1 \quad \mbox{$s_1,s_2,s_3$}\geq 1$$
 has $\binom{n-2}{2}$ solution.
I want to find any good formulae for the following form :

$$\sum\limits_{(s_1,s_2,s_3)}\prod_{i=1}^3\binom{s_i+s_{i-1}-1}{s_i}=?$$
where, $s_0=1$ and each $(s_1,s_2,s_3)$ is the solution of above equaiton.

I find that following:
$$n=4\Longrightarrow 1$$
$$n=5\Longrightarrow 5$$
$$n=6\Longrightarrow 18$$
$$n=7\Longrightarrow 57$$
$$n=8\Longrightarrow 169$$
$$n=9\Longrightarrow 502$$

>- My all attempts have failed