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
Explicit expressions are known for the Chebyshev polynomials, for example $$ U_n(x) = \sum_{k=0}^{\lfloor n/2 \rfloor} \binom{2k-n-1}{k} (2x)^{n-2k} $$ for $n \ge 1$.
The coefficient of $x^{n-2}$ is the $k=1$ term, that is $- (n-1) 2^{n-2} $ if $n \ge 2$, and zero otherwise.
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 $n\ge 1.$
