On the Unimodality of Convolutions of Sequences of Binomial Coefficients

We provide necessary and sufficient conditions on the unimodality of a convolution of two sequences of binomial coefficients preceded by a finite number of ones. These convolution sequences arise as rank sequences of posets of vertex-induced subtrees for a particular class of trees. The number of such trees whose poset of vertex-induced subgraphs containing the root is not rank unimodal is determined for a fixed number of vertices i.