I watched a youtube video (I cannot find it anymore) however, the author showed that the following binomial equation could be factored (If I remembered it correctly.)
$$\binom{n}{n+2k}=\binom{n}{n+k}\frac{1+(-1)^k}{2}$$
I cannot see how this was derived, for example I have tried the following:
$$\frac{n!}{(n+2k)!(n-(n+2k))!}$$
But cannot see how the above works.
What you probably saw was $$\sum_{k\ge 0} a_{2k} = \sum_{k\ge 0} a_k \frac{1+(-1)^k}{2},$$ which holds because $$\frac{1+(-1)^k}{2}= \begin{cases} 1&\text{if $2\mid k$},\\ 0 &\text{otherwise} \end{cases}$$