Concavity of entropy along binomial convolution

Motivated by a generalization of Sturm-Lott-Villani theory to discrete spaces and
by a conjecture stated by Shepp and Olkin about the entropy of sums of Bernoulli
random variables, we prove the concavity in t of the entropy of the convolution of a probability measure a, which has the law of a sum of independent Bernoulli variables,
by the binomial measure of parameters n and t.