The coefficient of a specific monomial of the following polynomial 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 is always zero. I use the superb conventions of Concrete Mathematics: Sums are over all integers unless otherwise indicated, and $\binom{n}{k}$ is $0$ if $k<0$ or $>n \geq 0$.
Expanding by the binomial theorem, 
$$f_{abc} = \sum_{i,j,k} (-1)^{i+j+k} \binom{2a+1}{i} \binom{2b+1}{j} \binom{2c+1}{k} x_1^{2a+1-i+k} x_2^{2b+1-j+i} x_3^{2c+1-k+j}.$$
We want
$$\begin{array}{r@{}c@{}lcr@{}c@{}l}
a+c+1 &=& 2a+1-i+k &\implies& c-k &=& a-i\\
a+b+1 &=& 2b+1-j+i &\implies& a-i &=& b-j\\
b+c+1 &=& 2c+1-k+j &\implies& b-j &=& c-k  \\
\end{array}$$
so $(i,j,k) = (a+r, b+r, c+r)$ for some $r$. We need to evaluate
$$\sum_r (-1)^{a+b+c+3r} \binom{2a+1}{a+r} \binom{2b+1}{b+r} \binom{2c+1}{c+r}.$$
Pair off the $r$ and $1-r$ terms to get
$$(-1)^{a+b+c} \sum_{r \geq 0} \left[ (-1)^{3r} \binom{2a+1}{a+r} \binom{2b+1}{b+r} \binom{2c+1}{c+r} + \right.$$
$$\phantom{(-1)^{a+b+c} \sum_{r \geq 1}} \left. (-1)^{3-3r} \binom{2a+1}{a+1-r} \binom{2b+1}{b+1-r} \binom{2c+1}{c+1-r} \right].$$
The two terms in the square brackets cancel, and the sum is zero.
