Consider the following series of polynomials in $t$:
$x+\frac1x=t=f_1(t)$
$x^2+\frac1{x^2}=t^2-2=f_2(t)$
$x^3+\frac1{x^3}=t^3-3t=f_3(t)$
$x^4+\frac1{x^4}=t^4-4t^2+2=f_4(t)$
$x^5+\frac1{x^5}=t^5-5t^3+5t=f_5(t)$
$x^6+\frac1{x^6}=t^6-6t^4+9t^2-2=f_6(t)$
$x^7+\frac1{x^7}=t^7-7t^5+14t^3-7t=f_7(t)$
$x^8+\frac1{x^8}=t^8-8t^6+20t^4-16t^2+2=f_8(t)$, etc.
They could be computed recursively using $x^n+\frac1{x^n}=(x^{n-1}+\frac1{x^{n-1}})(x+\frac1x)-(x^{n-2}+\frac 1{x^{n-2}})$.
Is there a name for these polynomials?
Is there a way to know their coefficients without computing them from previous polynomials in the series?
