Formula for a sum of product of binomials 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:
$$n=4\Longrightarrow 1$$
$$n=5\Longrightarrow 5$$
$$n=6\Longrightarrow 18$$
$$n=7\Longrightarrow 57$$
$$n=8\Longrightarrow 169$$
$$n=9\Longrightarrow 502$$

  
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*My all attempts have failed
  

 A: The generating function is $$ \sum_{s_1,s_2, s_3} {s_1 + s_2-1 \choose s_2} {s_2+ s_3-1 \choose s_3} x^{s_1+s_2+s_3}.$$
$$ \sum_{s_3} {s_2+ s_3-1 \choose s_3} x^{s_3} = \left( \frac{1}{1-x}\right)^{s_2}$$.
Then the sum over the $s_2$ variable is
$$ \sum_{s_2} {s_1 + s_2-1 \choose s_2}\left( \frac{x}{1-x}\right)^{s_2} = \left( \frac{1}{1- \frac{x}{1-x}} \right)^{s_1} = \left( \frac{1-x}{1-2x} \right)^{s_1}$$
and the sum over just the $s_1$ variable is
$$ \sum_{s_1} \left( \frac{x-x^2}{1-2x}\right)^{s_1} = \frac{1}{1- \frac{x-x^2}{1-2x}}=\frac{1-2x}{1-3x+x^2}.$$
However, we need to remove the terms when $s_1,s_2$ or $s_3$ is zero. Because of the binomial coefficients, if $s_1$ vanishes then $s_2$ vanishes and if $s_2$ vanishes then $s_3$ vanishes, so it suffices to remove the terms with $s_3=0$, which are
$$ \sum_{s_1,s_2} {s_1 + s_2-1 \choose s_2} x^{s_1+s_2}= \frac{1-x}{1-2x}$$ by the same logic. 
So the full generating function is $$  \frac{1-2x}{1-3x+x^2} - \frac{1-x}{1-2x}.$$
Your sum is then the coefficient of $x^{n-1}$ in this series. To get this, as EFinat-S suggests we may use partial fractions.
$$  \frac{1-2x}{1-3x+x^2} - \frac{1-x}{1-2x} =  \frac{- 2 + \sqrt{5} }{1- \frac{3 + \sqrt{5}}{2} x} + \frac{-2-\sqrt{5} }{1- \frac{3 - \sqrt{5}}{2} x} - \frac{1/2}{1-2x} - \frac{1}{2} $$
which will match the expression Carlo Beenaker gave.
Moreover, this will generalize to the analogue with $s_1,\dots,s_k$, giving a rational generating function. There is a straightforward enumerative interpretation, along the lines of the OEIS reference Carlo found as length $2(n-1)$ depth $k$ nested balanced parantheses expressions / plane trees / Dyck paths, which will thus be related to a column ofOEIS A080936.
A: Inspection of the Mathematica output
In: Table[Sum[Binomial[s2 + s1 - 1, s2]* Binomial[n - s1 - 2, n - 1 - s1 - s2], {s2, 1, n - 3}, {s1, 1,  n - 2 - s2}], {n, 4, 20}]
Out: {1, 5, 18, 57, 169, 482, 1341, 3669, 9922, 26609, 70929, 188226, 497845, 1313501, 3459042, 9096393, 23895673}

indicates it's the series OEIS A258109 
$$a_n=\frac{1}{5} 2^{-n-1} \left[\left(\sqrt{5}+5\right) \left(3-\sqrt{5}\right)^n-5\ 4^n-\left(\sqrt{5}-5\right) \left(\sqrt{5}+3\right)^n\right].$$
