I came across some polynomials which take the form
$$F_n(x)=\sum_{k=0}^{\lfloor n/2\rfloor}(-1)^k\binom{2k}{k}\binom{n-k}{k}x^{n-2k}.$$
I noticed that these look pretty similar to the series for the Legendre polynomials $$P_n(x)=\frac{1}{2^n}\sum_{k=0}^{\lfloor n/2\rfloor}(-1)^k\binom{2n-2k}{k}\binom{n}{k}x^{n-2k}$$ and a little bit like the Chebyshev polynomials of the second kind $$U_n(x)=\sum_{k=0}^{\lfloor n/2\rfloor}(-1)^k\binom{n-k}{k}(2x)^{n-2k}.$$ Is there some way I could work the polynomials $F_n(x)$ in terms of Legendre polynomials, or some other 'special' polynomials or functions? The main issue is the floor function making it slightly difficult for me to recognise it in terms of e.g. some hypergeometric function or other special polynomials.
Reference would also be a great answer in the case that these are some polynomials with their own name.