Let the real polynomial 
$$f\_{a,b,c}(x\_1,x\_2,x\_3)=(x\_1-x\_2)^{2a+1}(x\_2-x\_3)^{2b+1}(x\_3-x\_1)^{2c+1},$$
where $a,b,c$ are nonnegative integers.


Let $m\_{a,b,c}$ be the coefficient of the monomial $x\_1^{a+c+1}x\_2^{a+b+1}x\_3^{b+c+1}$ in the expansion of $f\_{a,b,c}(x\_1,x\_2,x\_3)$.


It is easy to see $m\_{a,b,c}=0$ when two of $a,b,c$ are equal. I want to ask whether $m\_{a,b,c}=0$ or not when $a,b,c$ are pairwise unequal.



The coefficient of a specific monomial of the following polynomial
This coefficient $L$ is a constant term of the Laurent polynomial $g(x\_1,x\_2,x\_3)=f(x\_1,x\_2,x\_3)/x\_1^{a+c+1}x\_2^{a+b+1}x\_3^{b+c+1}$, this guy $g$ satisfies $g(x\_1,x\_2,x\_3)=-g(1/x\_1,1/x\_2,1/x\_3)$, thus $L=-L$.


