  The notion of Ehrhart tensor polynomials, a natural generalization of the
Ehrhart polynomial of a lattice polytope, was recently introduced by Ludwig and
Silverstein. We initiate a study of their coefficients. In the vector and
matrix cases, we give Pick-type formulas in terms of triangulations of a
lattice polygon. As our main tool, we introduce $h^r$-tensor polynomials,
extending the notion of the Ehrhart $h^\ast$-polynomial, and, for matrices,
investigate their coefficients for positive semidefiniteness. In contrast to
the usual $h^\ast$-polynomial, the coefficients are in general not monotone
with respect to inclusion. Nevertheless, we are able to prove positive
semidefiniteness in dimension two. Based on computational results, we
conjecture positive semidefiniteness of the coefficients in higher dimensions.
Furthermore, we generalize Hibi's palindromic theorem for reflexive polytopes
to $h^r$-tensor polynomials and discuss possible future research directions.
