F. A Random Code Problem
combinatorics; dp; math; number theory; probabilities
You are given an integer array \(a_0, a_1, \dots, a_{n - 1}\), and an integer \(k\). You perform the following code with it:long long ans = 0; // create a 64-bit signed variable which is initially equal to 0for(int i = 1; i <= k; i++){ int idx = rnd.next(0, n - 1); // generate a random integer between 0 and n - 1, both inclusive // each integer from 0 to n - 1 has the same probability of being chosen ans += a[idx]; a[idx] -= (a[idx] % i);}Your task is to calculate the expected value of the variable ans after performing this code.Note that the input is generated according to special rules (see the input format section).
The only line contains six integers \(n\), \(a_0\), \(x\), \(y\), \(k\) and \(M\) (\(1 \le n \le 10^7\); \(1 \le a_0, x, y < M \le 998244353\); \(1 \le k \le 17\)).The array \(a\) in the input is constructed as follows: \(a_0\) is given in the input; for every \(i\) from \(1\) to \(n - 1\), the value of \(a_i\) can be calculated as \(a_i = (a_{i - 1} \cdot x + y) \bmod M\).
Let the expected value of the variable ans after performing the code be \(E\). It can be shown that \(E \cdot n^k\) is an integer. You have to output this integer modulo \(998244353\).
The array in the first example test is \([10, 35, 22]\). In the second example, it is \([15363, 1418543]\).