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*) = *c*_{0} + *c*_{1}*x* + *c*_{2}*x*^{2} + … + *c*_{n}*x*^{n} be a polynomial all of whose zeros are real and negative. Then the coefficient sequence *c*_{k} is strictly log concave.

This is Theorem 4.5.2 from *Generatingfunctionology*, available for download here.

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 kind, *s*(*n*, *k*), are log concave for fixed *n* and *k* ≥ 1. This because these numbers are the coefficients of *x*^{k} 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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