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sum exp times poly limit
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verify/poly/sum_exp_poly.test.cpp

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// @brief $\sum\limits_{i=0}^\infty r^i i^d$
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#define PROBLEM "https://judge.yosupo.jp/problem/sum_of_exponential_times_polynomial_limit"
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#pragma GCC optimize("Ofast,unroll-loops")
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#define CP_ALGO_MAXN 1 << 24
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#include "cp-algo/math/poly.hpp"
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#include <bits/stdc++.h>
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using namespace std;
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using namespace cp_algo::math;
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const int mod = 998244353;
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typedef modint<mod> base;
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typedef poly_t<base> polyn;
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base nCr(int n, int r) {
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return fact<base>(n) * rfact<base>(r) * rfact<base>(n-r);
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}
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polyn xm1k(size_t k) {
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polyn::Vector ans(k+1);
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for(int i = 0; i <= k; i++) {
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ans[i] = (k - i) & 1 ? -nCr(k, i) : nCr(k, i);
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}
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return ans;
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}
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void solve() {
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int r, d;
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cin >> r >> d;
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polyn num = polyn(bpow(base(1-r), d+1)) - bpow(base(r), d+1) * xm1k(d+1);
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polyn den = polyn(bpow(base(1-r), d+2)) - bpow(base(1-r), d+1) * r * (polyn::xk(1) - polyn(base(1)));
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polyn H = num / den;
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base ans = 0;
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base id[d+1];
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vector<int> lp(d+1);
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id[0] = bpow(base(0), d);
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id[1] = 1;
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for(int i = 2; i <= d; i++) {
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if(lp[i] == 0) {
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id[i] = bpow(base(i), d);
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for(int j = i; j <= d; j += i) {
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lp[j] = i;
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}
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} else {
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id[i] = id[lp[i]] * id[i / lp[i]];
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}
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}
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for(int i = 0; i <= d; i++) {
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ans += H[i] * id[i];
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}
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cout << ans << endl;
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}
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signed main() {
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//freopen("input.txt", "r", stdin);
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ios::sync_with_stdio(0);
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cin.tie(0);
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int t;
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t = 1;// cin >> t;
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while(t--) {
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solve();
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}
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}

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