  Recently, Glasby and Paseman considered the following sequence of binomial
sums $\{2^{-r}\sum_{i=0}^{r}\binom{m}{i}\}_{r=0}^{m}$ and showed that this
sequence is unimodal and attains its maximum value at
$r=\lfloor\frac{m}{3}\rfloor+1$ for
$m\in\mathbb{Z}_{\geq0}\setminus\{0,3,6,9,12\}$. They also analyzed the
asymptotic behavior of the maximum value of the sequence as $m$ approaches
infinity. In the present work, we generalize their results by considering the
sequence $\{(1+a)^{-r}\sum_{i=0}^{r}\binom{m}{i}a^{i}\}_{r=0}^{m}$ for integers
$a \geq 1$. We also consider a family of discrete probability distributions
that naturally arises from this sequence.

Combinatorics