Everything Is Generated In Equal Probability HDU-6595 期望DP

Solution

dp[n]=E+∑i=0n−1Cnidp[i]+dp[n]2ndp[n]=E+\frac{\sum\limits_{i=0}^{n-1}C_n^idp[i]+dp[n]}{2^n}dp[n]=E+2ni=0∑n−1Cnidp[i]+dp[n]

E=n(n−1)4E=\frac{n(n-1)}{4}E=4n(n−1)

dp[n]=n(n−1)4+∑i=0n−1Cnidp[i]2n1−12ndp[n]=\frac{\frac{n(n-1)}{4}+\frac{\sum\limits_{i=0}^{n-1}C_n^idp[i]}{2^n}}{1-\frac{1}{2^n}}dp[n]=1−2n14n(n−1)+2ni=0∑n−1Cnidp[i]

res=∑i=0ndp[i]nres=\frac{\sum\limits_{i=0}^{n}dp[i]}{n}res=ni=0∑ndp[i]

Code

/*SiberianSquirrel*/
/*CuteKiloFish*/
#include <bits/stdc++.h>
using namespace std;typedef long long ll;const ll MOD = 998244353;int dp[3010] = {0, 0};
ll quick_pow(ll ans, ll p, ll res = 1) {for(; p; p >>= 1, ans = ans * ans % MOD)if(p & 1) res = res * ans % MOD;return res % MOD;
}
ll inv(ll ans) {return quick_pow(ans , MOD - 2);
}
ll inv4 = inv(4);
ll fac[3010] = {1, 1};
ll C[3010][3010];
void solve() {for(ll i = 2; i <= 3000; ++ i) {fac[i] = fac[i - 1] * i % MOD;}for(int i = 1; i <= 3000; ++ i) {C[i][0] = C[i][i] = 1;for(int j = 1; j < i; ++ j) {C[i][j] = (C[i - 1][j] + C[i - 1][j - 1]) % MOD;}}for(ll i = 2; i <= 3000; ++ i) {ll res = i * (i - 1) % MOD * inv4 % MOD;ll ans = 0;for(ll j = 0; j <= i - 1; ++ j) {ans = (ans + C[i][j] * dp[j] % MOD) % MOD;}ll invn2 = inv(quick_pow(2, i));ans = ans * invn2 % MOD;ll temp = inv((1 - invn2 + MOD) % MOD);dp[i] = (res + ans) % MOD * temp % MOD;}int n; while (cin >> n) {ll res = 0;for(int i = 0; i <= n; ++ i) {res = (res + dp[i]) % MOD;}res = res * inv(n) % MOD;cout << res << endl;}
}signed main() {ios_base::sync_with_stdio(false);cin.tie(0);cout.tie(0);
#ifdef ACM_LOCALfreopen("input", "r", stdin);freopen("output", "w", stdout);signed test_index_for_debug = 1;char acm_local_for_debug = 0;do {if (acm_local_for_debug == '$') exit(0);if (test_index_for_debug > 20)throw runtime_error("Check the stdin!!!");auto start_clock_for_debug = clock();solve();auto end_clock_for_debug = clock();cout << "Test " << test_index_for_debug << " successful" << endl;cerr << "Test " << test_index_for_debug++ << " Run Time: "<< double(end_clock_for_debug - start_clock_for_debug) / CLOCKS_PER_SEC << "s" << endl;cout << "--------------------------------------------------" << endl;} while (cin >> acm_local_for_debug && cin.putback(acm_local_for_debug));
#elsesolve();
#endifreturn 0;
}

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Everything Is Generated In Equal Probability HDU-6595 期望DP

Solution

Code

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