<s>[INST] Finding pattern in a sequence of polynomials So I have a set of polynomials, with variables $n_0, n_1, n_2$. I would like to figure out the general formula for these polynomials given the number.
$$f(2)  = n_{0} + 2 n_{1} + n_{2}$$
$$f(3)  = n_{0}^{3} + 3 n_{0} n_{1}^{2} + 3 n_{1}^{2} n_{2} + n_{2}^{3}$$
$$f(4)  = n_{0}^{6} + 4 n_{0}^{3} n_{1}^{3} + 6 n_{0} n_{1}^{4} n_{2} + 4 n_{1}^{3}     n_{2}^{3} + n_{2}^{6}$$
$$f(5)  = n_{0}^{10} + 5 n_{0}^{6} n_{1}^{4} + 10 n_{0}^{3} n_{1}^{6} n_{2} + 10 n_{    0} n_{1}^{6} n_{2}^{3} + 5 n_{1}^{4} n_{2}^{6} + n_{2}^{10}$$
$$f(6)  = n_{0}^{15} + 6 n_{0}^{10} n_{1}^{5} + 15 n_{0}^{6} n_{1}^{8} n_{2} + 20 n    _{0}^{3} n_{1}^{9} n_{2}^{3} + 15 n_{0} n_{1}^{8} n_{2}^{6} + 6 n_{1}^{5} n_{2}^{10} + n_{2}^{15}$$
$$f(7)  = n_{0}^{21} + 7 n_{0}^{15} n_{1}^{6} + 21 n_{0}^{10} n_{1}^{10} n_{2} +     35 n_{0}^{6} n_{1}^{12} n_{2}^{3} + 35 n_{0}^{3} n_{1}^{12} n_{2}^{6} + 21 n_{0} n_{1}^{10} n_{2}   ^{10} + 7 n_{1}^{6} n_{2}^{15} + n_{2}^{21}$$
As you can see that the coefficients are given by binomial expansion but the variables multiply and have degrees in a non-trivial manner. Is there a pattern to degree and combinations of variables, can a closed-form in the number of constants (i.e $1,2,3,4 ..$) be achieved ?? I can post a larger list of polynomials if needs be. 
 [/INST] I think it is
$$
f(k)=\sum\limits_{j = 0}^{k} \binom{k}{j}n_0^{\frac{(k - j)(k - j - 1)}{2}} n_1^{(k-j)j} n_2^{\frac{j(j - 1)}{2}} 
$$
for $k\geq 2$.
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