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# Log Concave Coefficients

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A few days ago I wrote about the rise and fall of binomial coefficients. There I gave a proof that binomial coefficients are log-concave, and so a local maximum has to be a global maximum.

Here I’ll give a one-line proof of the same result, taking advantage of the following useful theorem.

Let p(x) = c0 + c1x + c2x2 + … + cnxn be a polynomial all of whose zeros are real and negative. Then the coefficient sequence ck is strictly log concave.

Now for the promised one-line proof. Binomial coefficients are the coefficients of (x + 1)n, which is clearly a polynomial with only real negative roots.

The same theorem shows that Stirling numbers of the first kinds(nk), are log concave for fixed n and k ≥ 1. This because these numbers are the coefficients of xk in

(x + 1)(x + 2) … (x + n – 1).

The theorem can also show that Stirling numbers of the second kind are log-concave, but in that case the generating polynomial is not so easy to write out.

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