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.
Categories: math.CO