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So, I'm wondering if the relationship $$x^n=a_1x^{n-1}+a_2x^{n-2}+...+a_{n-1}x+a_n$$ exists, where the value of the left and right side are equal for all integer values of x greater than zero up to the integer n.

I am looking for being able to find the coefficients that makes this true.

One example I can think of is for n=3 then $$x^3=6x^2-11x+6$$ which is true when x=1, 2, or 3 In this case I care about the fact that the coefficients are 6, -11, and 6

Will Johnson
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  • I don't quite understand what your question. Do you ask whether there exist coefficients $a_i$ such that the equality holds foe every integer $x<n$? – boaz Nov 01 '16 at 17:09
  • What you are looking for is "interpolation polynomial". More precisely, you are trying to interpolate the function $x\mapsto x^n$ on the points $1,2,\dots,n$ with a polynomial of degree $<n$. However, in this precise case, Robert's method gives you the coefficients in the fastest possible way :-) – Nicolas FRANCOIS Nov 01 '16 at 17:21

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The difference between the sides of your equation is a monic polynomial of degree $n$, and you're asking whether it can have roots $1, 2, \ldots, n$. Yes, it can: $(x-1)(x-2)\ldots(x-n)$.

EDIT: You can write the coefficients using the elementary symmetric functions of the roots.

Robert Israel
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