  Using the following $_4F_3$ transformation formula $$
\sum_{k=0}^{n}{-x-1\choose k}^2{x\choose n-k}^2=\sum_{k=0}^{n}{n+k\choose
2k}{2k\choose k}^2{x+k\choose 2k}, $$ which can be proved by Zeilberger's
algorithm, we confirm some special cases of a recent conjecture of Z.-W. Sun on
integer-valued polynomials.
