I have the following summation: $$H=\sum^{n}_{i=0}t_i\log \binom{n+1}{i}-\log(t_i !)$$ where $$t_i=\epsilon\Big(\binom{n}{i}-\binom{n-k}{i-k}+\binom{n-k}{i-1}\Big)$$ where $\epsilon$ and $k$ are some positive constants, also $k<<n$. I wonder if there is an asymptotic form for $n\rightarrow \infty$ can be obtained?
NOTE: $t_i$ is an integer, and we would evaluate $t_i$ for $i\geq k$.