During my research I got to the point when I need to find
$$ \arg \max_w \left( (n-w) \sum_{j=0}^d \binom{w}{j} \binom{2^r - (j+1) 2^{r-j-1}-2}{t} \right) $$
with respect to $w$ only (i.e. $d$, $n$, $r$ and $t$ are considered as integer constants).
Again, I don't need the value of the maximized expression, I need only $w$ which maximizes the expression.
I could see some obvious conditions on the parameters (like $d \leqslant w$) but nothing interesting.
Combinatorics and sums are not my field so I don't really know what to do with this and what I can do in principle.
Approximations would be fine too, provided they are not too rough. Any hints or links would be also appreciated.
It seems that many terms are irrelevant in terms of optimization. Thus, we can rewrite the problem as $$ \arg\max_w \left( (n-w)\sum_{j=0}^d {w\choose j}\alpha_j\right) $$ Since the combination numbers can be expressed using gamma function, we can write $$ (n-w)\sum_{j=0}^d {w\choose j}\alpha_j= (n-w)\sum_{j=0}^d \frac{w!}{(w-j)!j!}\alpha_j=(n-w)\sum_{j=0}^d \frac{\Gamma(w+1)}{\Gamma(w-j+1)\Gamma(j+1)}\alpha_j $$ which is continuous in $w$ and can be subject of differentiation and standard optimization methods and tools.