Suppose $c,t$ are such that, $0< c< 1$ constant and $cn\leq t \leq n$. I want to have an estimation of

$\sum _{i=0}^{cn} {cn\choose {i}}{(1-c)n \choose t-i} 2^{t-i}$

when n goes to infinity.

Can I bound it by $2^{c'n}$ for some $0<c'<log_2(3)$?

I have no idea to do that.Is there any hint?

Thanks!