This is answered affirmatively by Odlyzko and Richmond, On the unimodality of high convolutions of discrete distributions, Annals of probability (1985) 299--306: all sufficiently large powers of the polynomial (with positive coefficients and no gaps) are strongly unimodal, that is, the coefficients form a log convex sequence. The proof uses estimates of contour integrals over circles of just the right radius.

This is answered affirmatively by Odlyzko and Richmond, On the unimodality of high convolutions of discrete distributions, Annals of probability (1985) 299--306: all sufficiently large powers of the polynomial (with positive coefficients and no gaps) are strongly unimodal, that is, the coefficients form a log concave sequence. The proof uses estimates of contour integrals over circles of just the right radius.