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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.

SagarM
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1 Answers1

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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 $k\geq 2$.

Gary
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  • I have not checked it yet, but I am quite amazed already, Can you guide me as to how you reached on this expression? – SagarM Mar 01 '20 at 21:57
  • Thanks a lot, your answer works. But any hint into how you found it would be immensely helpful. – SagarM Mar 01 '20 at 23:04
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    Well, I just had to look at the polynomials long enough to figure out the patter in the powers. – Gary Mar 02 '20 at 06:22
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    I have simplified the formula a bit. – Gary Mar 02 '20 at 07:10
  • this has been really helpful. I know, you could do it in your head, but is there any book with these kinds of exercises or at least what field of mathematics would deal with this kind of problems.(Where I could have more complicated patterns to detect.) – SagarM Mar 02 '20 at 09:57
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    I believe it is just a question of experience. I cannot recommend any particular book. – Gary Mar 02 '20 at 10:49