I checked for myself in Wolfram some initial values for n
and it was ok however I would like to see a full proof with induction being enough for me
(When proving the induction step, you will probably need to refer to the definition of recursion
except that here we have second-order linear...
Not long ago, I derived the formula for Chebyshev polynomials
$$T_{n}\left( x\right)= \sum_{k=0}^{\lfloor \frac{n}{2} \rfloor}{n \choose 2k}x^{n-2k}\left( x^2-1\right)^{k}$$
How to extract the coefficients of this polynomial of degree n ?
I tried using Newton's binomial but got a double sum...