  We express a weighted generalization of the Delannoy numbers in terms of
shifted Jacobi polynomials. A specialization of our formulas extends a relation
between the central Delannoy numbers and Legendre polynomials, observed over 50
years ago, to all Delannoy numbers and certain Jacobi polynomials. Another
specialization provides a weighted lattice path enumeration model for shifted
Jacobi polynomials, we use this to present a new combinatorial proof of the
orthogonality of Jacobi polynomials with natural number parameters. The proof
relates the orthogonality of these polynomials to the orthogonality of
(generalized) Laguerre polynomials, as they arise in the theory of rook
polynomials. We provide a combinatorial proof for the orthogonality of certain
Romanovski-Jacobi polynomials with zero first parameter and negative integer
second parameter, considered as an initial segment of the list of similarly
transformed Jacobi polynomials with the same parameters. We observe that for an
odd second parameter one more polynomial may be added to this finite orthogonal
polynomial sequence than what was predicted by a classical result of Askey and
Romanovski. The remaining transformed Jacobi polynomials in the sequence are
either equal to the already listed Romanovski-Jacobi polynomials or monomial
multiples of similarly transformed Jacobi polynomials with a positive second
parameter. We provide expressions for an analogous weighted generalization of
the Schr\"oder numbers in terms of the Jacobi polynomials, and use this model,
together with a result of Mansour and Sun, to express the Narayana polynomials
in terms of shifted Jacobi polynomials.
