Let be a positive integer. The binomial coefficient is called a *central binomial coefficient*. Several bounds for these coefficients are known and the more advanced ones are commonly derived from Stirling’s formula for factorials or from Wallis’ product for . This post presents an alternative and self-contained elementary proof of the bounds

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Define a function by

and take . From and it follows that

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So is non-increasing on this interval and therefore or

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This inequality clearly also holds at the endpoint and since both sides are symmetric in it holds throughout the interval . (It suggests that closely resembles a normal distribution on this interval.) Integration of the inequality above leads to

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Both the far left and right hand sides of these inequality can be computed explicitly. Starting with the right hand side let

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Then

And so . To evaluate the left hand side let

By explicit computation we find and . The other values can be found by a recursive relation that follows from partial integration. For positive we have

and therefore the recursion . Using this recursion one can check that the even and odd entries of the sequence are given respectively by

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Putting all results sofar together we find for positive even integers

and for odd integers

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