The wikipedia page of central binomial coefficient mentions an upper bound of the central binomial coefficient $$\binom{2n}{n}=\frac{4^n}{\sqrt{\pi n}}(1-\frac{c_n}{n}),\quad c_n\in (1/9,1/8),\forall n\ge 1\,.$$ However I was unable to find any reference or proof of this inequality. It would be great if anyone would show me a reference for it so that I can learn the techniques. Thank you.
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There is a very good paper at https://www.emis.de/journals/JIPAM/images/043_00_JIPAM/043_00.pdf – Claude Leibovici Oct 02 '19 at 04:16
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Have a look at @robjohn's answer in this question : https://math.stackexchange.com/questions/58560/elementary-central-binomial-coefficient-estimates – Claude Leibovici Oct 02 '19 at 04:22
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@ClaudeLeibovici thank you for the great references! – Alex Oct 02 '19 at 04:30
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You are very welcome ! The second one I sent is very nice (simple when you know the answer !) as all work done by @robjohn, a very eminent user in this site. Cheers. – Claude Leibovici Oct 02 '19 at 04:46