I have the following Binomial expansion which I want to bound tightly. I have been able to bound but it is very loose. My method is as follows :-
$2\sum_{i=1}^{k/2} {n \choose n/2 +i} + {n \choose n/2}=2\left[ {n \choose n/2 +1} +{n \choose n/2 +2} \ldots +{n \choose n/2 +k/2} \right] + {n \choose n/2}$
$=2\left[ {n+1 \choose n/2 +2} +{n+1 \choose n/2 +4} \ldots +{n+1 \choose n/2 +k/2} \right] + {n \choose n/2}~~~~~$ (Combining 2 terms using ${n \choose k} + {n \choose k-1}={n+1 \choose k}$)
$\ge 2 \left[{n+1 \choose n/2 +3} + {n+1 \choose n/2 +4 } \ldots {n+1 \choose n/2 +k/2 -1} + {n+1 \choose n/2 +k/2} \right] + {n \choose n/2}~~~~~~~$ (Using the inequality ${n+1 \choose n/2+k+1}\ge {n+1 \choose n/2+k}$)
$\ge 2{n+\log k \choose n/2 + k/2} + {n \choose n/2}$
Is it possible to have a tighter bound for this summation than what I obtained?