Sequence closed expression or others

Question

What are closed expression or any other expression (involving integrals, specials functions...) for

$\sum_{k=0}^{n}(n-2k)^t\frac{n!}{k!(n-k)!}$

where $t>0$ integer Thank you

Answer 1

Define $$f\left(x\right)=\left(2\cosh\left(x\right)\right)^{n}=\left(e^{x}+e^{-x}\right)^{n}.$$ By binomial theorem$$f\left(x\right)=\sum_{k=0}^{n}\dbinom{n}{k}e^{\left(n-k\right)x}e^{-kx}=\sum_{k=0}^{n}\dbinom{n}{k}e^{\left(n-2k\right)x}$$ hence$$\frac{d^{t}f}{dx^{t}}\left(x\right)=\sum_{k=0}^{n}\dbinom{n}{k}\left(n-2k\right)^{t}e^{\left(n-2k\right)x}$$ then$$\sum_{k=0}^{n}\dbinom{n}{k}\left(n-2k\right)^{t}=\frac{d^{t}f}{dx^{t}}\left(0\right).$$