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An interesting inequality on binomial coefficients
Johnson-Newman-Winston [1]:
For $n\geq 13$,
\[
\sum_{i=0}^{h-1}\binom{n}{i} > \binom{n}{h}
\]
if and only if
\[
h\geq \left\lfloor\frac{n}{3} \right\rfloor +2,
\]
where $\lfloor x\rfloor$ is the largest integer less than or equal to $x$.
Reference:
- E.L. Johnson, D. Newman, K. Winston, An inequality on binomial coefficients,
In Algorithmic Aspects of Combinatorics , Annals of Discrete Math., vol 2 (1978) 155-159. Local copy is here
S. Jukna, December, 2015