Properties of Motzkin triangle and t-generalized Motzkin sequences

We consider the Motzkin triangle as the zero-free part of a well-defined plane array. The right diagonal leg of the triangle is the Motzkin sequence, which satisfies a second order linear recurrence with linear polynomial coefficients. We extend this relation to the parallel diagonals to the line of Motzkin sequence. More generally, we prove the existence of a recursive formula for the formation of three arbitrary elements in the triangle, and construct the corresponding formulae for three connected entries, among them diagonal triples, of twenty possible formations. These recursive formulae have bivariate polynomial coefficients of higher order. We describe the columns of the Motzkin triangle as polynomial values, and reveal nice non-trivial factorization properties of these polynomials. The results essentially depend on an initially derived recursive rule of three consecutive horizontal elements provided by the definition of the triangle, and on a construction method which creates recurrence rules for other structures.