  We develop a monotonicity formula for solutions of the fractional Toda system
$$ (-\Delta)^s f_\alpha = e^{-(f_{\alpha+1}-f_\alpha)} -
e^{-(f_\alpha-f_{\alpha-1})} \quad \text{in} \ \ \mathbb R^n,$$ when $0<s<1$,
$\alpha=1,\cdots,Q$, $f_0=-\infty$, $f_{Q+1}=\infty$ and $Q \ge2$ is the number
of equations in this system. We then apply this formula, technical integral
estimates, classification of stable homogeneous solutions, and blow-down
analysis arguments to establish Liouville type theorems for finite Morse index
(and stable) solutions of the above system when $n > 2s$ and $$
\dfrac{\Gamma(\frac{n}{2})\Gamma(1+s)}{\Gamma(\frac{n-2s}{2})} \frac{Q(Q-1)}{2}
> \frac{ \Gamma(\frac{n+2s}{4})^2 }{ \Gamma(\frac{n-2s}{4})^2} . $$ Here,
$\Gamma$ is the Gamma function. When $Q=2$, the above equation is the classical
(fractional) Gelfand-Liouville equation.
