Given $x+\dfrac1x=n$, I derived several expressions in terms of $n$ to solve for $x^k+\dfrac1{x^k}$ and put them in a chart as shown below. My questions is how are the coefficients of these polynomials related? I have observed that the first coefficient is $1$, the second coefficient is $-k$, and the signs always alternate.
If $x + 1/x = n$, writing $x = \exp(it)$ we have $\cos(t) = n/2$. Then $x^k + 1/x^k = \exp(ikt) + \exp(-ikt) = 2 \cos(kt) = 2 T_k(n/2)$ where $T_k$ is the $k$'th Chebyshev polynomial of the first kind. Thus for $k=11$, $$T_{11}(x) = 1024\,{x}^{11}-2816\,{x}^{9}+2816\,{x}^{7}-1232\,{x}^{5}+220\,{x}^{3}- 11\,x$$ so you get $$ 2 T_{11}(n/2) = {n}^{11}-11\,{n}^{9}+44\,{n}^{7}-77\,{n}^{5}+55\,{n}^{3}-11\,n$$