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.
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 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.
This reminds me so much of results involving Newton Polygons in that they also relate the heights of coefficients to the factorization of polynomials. However, in that case the heights are not the normal heights of this results but the p-adic heights. So I don’t think that machinery can prove your result.