This is probably rather trivial, but anything that's combinatorics related messes me up. Just to check: are the following summations identical?
$$\sum_{k_1+k_2+k_3 = n} a^{k_1} b^{k_2} c^{k_3} \quad \text{and} \quad \sum_{0\leq i \leq j \leq n} a^i b^{j-i} c^{n-j}$$
where $k_1,k_2,k_3 \geq 0$. I believe both summations have the same number of terms, namely $\frac{(n+1)(n+2)}{2}$, but I'd like to check.