# Asymptotic approximation of a sum involving log and binomial coefficient

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 $i\geq k$.

• Is your question: "$t_n \sim\, ?\,$ for large $n$" ? Or "$H \sim\, ?\,$ for large $n$" ? – user90369 Aug 28 '18 at 14:58
• "H" the whole equation. However it would be interesting to look at both as you mentioned. – William Aug 28 '18 at 15:23