<s>[INST] Reducing $\frac{a^n(b-c) - b^n(a-c) + c^n(a-b)}{(a-b)(a-c)(b-c)}$ for $n>2$ 
Simplify
$$\frac{a^n(b-c) - b^n(a-c) + c^n(a-b)}{(a-b)(a-c)(b-c)}$$ for $n>2$.

The answer is $(a+b+c)^n$, but I can't seen to get it.
Can someone help me?
Thanks
 [/INST] The answer is neither $(a+b+c)^n$ nor $(a-b+c)^n$.
As can be found here and  here , we have that
$$
\frac{(b-c)\times a^n - (a-c)\times b^n + (a-b)\times c^n}{(a-b)(a-c)(b-c)}\\
 = \frac{a^n}{(a-b)(a-c)}+\frac{b^n}{(b-a)(b-c)}+\frac{c^n}{(c-a)(c-b)} = 
\displaystyle\sum_{\substack{n_1, n_2, n_3 \geqslant 0 \\ n_1 + n_2 + n_3 = n - 2}}a^{n_1}b^{n_2}c^{n_3}
$$
Some examples:
$$P_2 = 1 \\
P_3 = a + b  +c \\
P_4 = a^2 + b^2 + c^2 + ab + bc + ac 
$$
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