General formula for ${n \choose r} + {n-2 \choose r} + {n-4 \choose r} + ... + {n-2k \choose r}$

Question

I'm trying to figure out a general formula where $n$ decrements by 2 until $n - 2k = r$.

A more eloquent way of writing it would be

$\sum\limits_{i = 0}^{k} {r + 2i \choose r}$

Answer 1

Hint Perhaps this can help.

Consider the following $$(1+x)^{r+2k}+(1+x)^{r+2k-2}+\dotsb +(1+x)^{r}$$ Your expression is the coefficient of $x^r$ in this polynomial. Observe that this polynomial is a geometric series, so you want to find the coefficient of $x^r$ in $$\frac{(1+x)^{r}(1-(1+x)^{2(k+1)})}{1-(1+x)^2}$$

Although I'm not sure if this will lead to an elegant answer.