  Recently, Z.-W. Sun introduced two kinds of polynomials related to the
Delannoy numbers, and proved some supercongruences on sums involving those
polynomials. We deduce new summation formulas for squares of those polynomials
and use them to prove that certain rational sums involving even powers of those
polynomials are integers whenever they are evaluated at integers. This confirms
two conjectures of Z.-W. Sun. We also conjecture that many of these results
have neat $q$-analogues.
