In arXiv:math/0407326 Emeric Deutsch and Bruce Sagan proved that the Delannoy numbers $$D_n=\sum{\binom{n}{k}}^22^k $$
satisfy $D_n\not\equiv0\mod3$ if and only if $n=2m$ with $\binom{2m}{m}\not\equiv 0\mod{(3)}.$
It seems that this can be generalized to: $$\sum{\binom{n}{k}}^2t^k\not\equiv0\mod{(t+1)}$$ if and only if $$n=2m$$ with $$\binom{2m}{m}\not\equiv 0\mod{(t+1)}.$$
My question is: Is this result known and if so where is it published?
Sorry, my question was a generalization of a rather trivial reformulation of the cited result. The congruence follows from the identity $\sum{\binom{n}{k}}^2t^k =\sum\binom{2j}{j}\binom{n}{2j}t^j(1+t)^{n-2j}.$