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 :

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

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