Speeding up the Rummikub algorithm – explanation required

Regarding this question: Rummikub algorithm.

I was reading the first part of the solution in the posted answer (specifically, when there are no jokers involved, all tiles are distinct and only four colours are involved). Then, I reached the part in which the says that the algorithm runs in $ O(ABCD(A+B+C+D))$ time, which is easy to determine why.

However, he the goes on to saying that we can speed up the algorithm so as to run in $ O(ABCD)$ time by changing "the recurrence to ensure this occurs only once while maintaining correctness, which leads to $ O(1)$ time for every ‘cell’ in the DP-table".

My problem is: I do not see how this can be done. I have tried playing around a bit with the recurrence, but I do not see how it can be modified, or what else we should keep track of so that we can speed up the time.