A11664. 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.

The first line contains two integers, $N$ ( $1<=N<=10^{5}$ ) — the maximum number written on the blackboard — and $M$ ( $0<=M<=10^{18}$ ) — the number of steps to perform.

The second line contains $N+1$ integers $P_{0},P_{1},...,P_{N}$ ( $0<=P_{i}<998244353$ ), where $P_{i}$ describes the probability that the starting number is $i$ . We can express this probability as irreducible fraction $P/Q$ , then . It is guaranteed that the sum of all $P_{i}$ s equals $1$ (modulo $998244353$ ).

Output a single line of $N+1$ integers, where $R_{i}$ is the probability that the final number after $M$ steps is $i$ . It can be proven that the probability may always be expressed as an irreducible fraction $P/Q$ . You are asked to output .

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.