Is there a way to find expansion of powers of multinomials without any coefficients?

Question

For example, $(a + b + c)^3 = a^3 + b^3 + c^3 + 3ab^2 + 3ac^2 + 3a^2b + 3a^2c + 3bc^2 + 3b^2c + 6abc$ Knowing the value of a, b and c, is there a way to find this without the coefficients i.e. $a^3 + b^3 + c^3 + ab^2 + ac^2 + a^2b + a^2c + bc^2 + b^2c + abc$ using previous values or some other way?

Answer 1

The answer to this question gives the formula you are looking for ($m=3$).

$$\frac{a^{m+2}}{(a-b)(a-c)}+\frac{b^{m+2}}{(b-a)(b-c)}+\frac{c^{m+2}}{(c-a)(c-b)}$$