Consider the function $f(x) = (1+x)^n = \sum\limits_{i=0}^n \binom{n}{i}x^i, x > 1$ and $h(x) = (1-x)^n = \sum\limits_{i=0}^n \binom{n}{i}(-x)^i, x > 1$, we are interested in the $x[f(x) + h(x)] + f(x) - h(x)$.
$f(x) + h(x)$ is 2* sum of even items of $f(x)$ while $f(x) - h(x)$ is 2* sum of odd items of $f(x)$. I am wondering if there is any simple form for the sum of even items or odd items?