CF923E Perpetual Subtraction

There is a number $ x $ initially written on a blackboard. You repeat the following action a fixed amount of times: 1. take the number $ x $ currently written on a blackboard and erase it 2. select an integer uniformly at random from the range $ [0,x] $ inclusive, and write it on the blackboard Determine the distribution of final number given the distribution of initial number and the number of steps.

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In the first case, we start with number 2. After one step, it will be 0, 1 or 2 with probability 1/3 each. In the second case, the number will remain 2 with probability 1/9. With probability 1/9 it stays 2 in the first round and changes to 1 in the next, and with probability 1/6 changes to 1 in the first round and stays in the second. In all other cases the final integer is 0.