I'm currently playing with some sequence and series studying asymptotic properties and stuff and I'd need to know if it exists a closed form for:
$$s^k_n:=\sum_{l=0}^n(-1)^l{{l+k}\choose k}$$
for $k\in \mathbb{N}$. so far I know:
$$s^0_n=\left[ \frac n2 \right]$$
$$s^1_n=\frac {1+(2n+3)(-1)^n}4$$
any help (ways to do it by induction, recursive formula, asymptotic behavior, source, etc) is greatly appreciated.
$$s^k_n=2^{-k-1}+(-1)^n\left(^{n+k+1}_{\ \ \ \ k}\right)hypergeom([1,n+k+2],[n+2],-1)$$
$$s^k_\infty=2^{-k-1}$$