Stanley's conjectures on the Stern poset

The Stern poset S is a graded infinite naturally associated with Stern's triangle, which was defined by Stanley, in analogy Pascal's triangle. Stanley noted that every interval distributive lattice. Let Pn denote the unique (up to isomorphism) for set of its order ideals, ordered inclusion, isomorphic from element row 0 triangle n-th r sufficiently large r. For n?1 letLn(q)=2?(?k=12n?1APk(q))+AP2n(q), where AP(q) represents corresponding P-Eulerian polynomial. all conjectured Ln(q) has only real zeros and L4n+1(q) divisible L2n(q). In this paper we obtain simple recurrence relation satisfied affirmatively solve Stanley's conjectures.