Geodesic
On the maximum of the weighted binomial sum \(2^{-r}\sum_{i=0}^r\binom{m}{i}\)
Stephen Glasby1 ; Gerhard Paseman2
1The University of Western Australia
2Sheperd Systems
The electronic journal of combinatorics, Tome 29 (2022) no. 2
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The weighted binomial sum $f_m(r)=2^{-r}\sum_{i=0}^r\binom{m}{i}$ arises in coding theory and information theory. We prove that, for $m\not\in\{0,3,6,9,12\}$, the maximum value of $f_m(r)$ with $0\le r\le m$ occurs when $r=\lfloor m/3\rfloor+1$. We also show this maximum value is asymptotic to $\frac{3}{\sqrt{{\pi}m}}\left(\frac{3}{2}\right)^m$ as $m\to\infty$.
DOI : 10.37236/10751
Classification : 05A10, 11B65
Mots-clés : binomial coefficients, Stirling number

Stephen Glasby  1   ; Gerhard Paseman  2

1 The University of Western Australia
2 Sheperd Systems 
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Stephen Glasby; Gerhard Paseman. On the maximum of the weighted binomial sum \(2^{-r}\sum_{i=0}^r\binom{m}{i}\). The electronic journal of combinatorics, Tome 29 (2022) no. 2. doi: 10.37236/10751

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