In the course of solving some optimization, I encountered the following sum:
$\sum_{l=0}^j (-1)^l 3^{j-l}\binom{j}{l}\sum_{k=0}^n \binom{n-j}{k-l}=2^n, \quad j\in\left\{0,1,\dots,n\right\}$.
I am interested in obtaining closed-form lower and upper bounds on the partial sum of this kind, i.e.
$\sum_{l=0}^j (-1)^l 3^{j-l}\binom{j}{l}\sum_{k=0}^{\alpha n} \binom{n-j}{k-l}, \quad j\in\left\{0,1,\dots,n\right\}$.
A result for $0.5 < \alpha < 1$ fixed would still be useful.
I have been trying to deal with it for a longer period of time, look for some combinatoric identities etc. but the alternating terms and the partial sum of a binomial coefficient make it really complicated. I do not specialize in this kind of stuff, though, so I decided to look for help here.
Any hints are welcome.
Thanks.