What is the coefficient of the power $x^{n-2}$ in a Chebyshev polynomial? Let us consider the second kind Chebyshev polynomial over the positive integers $U_{n+1}(x) = 2xU_n(x) - U_{n-1}(x)$ with $n>1$ is a positive integer.
I know that the leading term of $U_n(x)$ is $2^{n}$ and it's associated with the power $x^{n}$.
My question is:
What is the coefficient of the power $x^{n-2}$?
I find the formula (17) in this link: https://mathworld.wolfram.com/ChebyshevPolynomialoftheSecondKind.html

You can get the result directly from the recurrence relation. Let $$U_n(x)=2^nx^n -c_nx^{n-2}+\ldots $$  As $U_0=1$ and $U_1=2x,$ we get $c_0=c_1=0.$ Next making use of the recurrence relation gives
$$2^{n+1}x^{n+1}-c_{n+1}x^{n-1}+\ldots = (2^{n+1}x^{n+1}-2c_nx^{n-1}+\ldots )-(2^{n-1}x^{n-1}-c_{n-1}x^{n-3}+\ldots ) $$
We equate the coefficients at $x^{n-1}$ and divide by $2^{n+1}$ to  get
$$2^{-n-1}c_{n+1}=2^{-n}c_n+{1\over 4}, \qquad n\ge 1.$$
Therefore $$2^{-n}c_n={n-1\over 4}+c_1={n-1\over 4}.$$
Hence $c_n=(n-1)2^{n-2}$ for $n\ge 1.$
