結果
| 問題 |
No.2005 Sum of Power Sums
|
| コンテスト | |
| ユーザー |
 sigma425
|
| 提出日時 | 2022-07-10 00:44:18 |
| 言語 | C++17 (gcc 13.3.0 + boost 1.87.0) |
| 結果 |
AC
|
| 実行時間 | 607 ms / 2,000 ms |
| コード長 | 17,109 bytes |
| コンパイル時間 | 2,750 ms |
| コンパイル使用メモリ | 215,924 KB |
| 最終ジャッジ日時 | 2025-01-30 06:01:22 |
|
ジャッジサーバーID （参考情報） |
judge5 / judge3 |
（要ログイン）
| ファイルパターン | 結果 |
|---|---|
| sample | AC * 3 |
| other | AC * 18 |
ソースコード
#include <bits/stdc++.h>
using namespace std;
using ll = long long;
using uint = unsigned int;
using ull = unsigned long long;
#define rep(i,n) for(int i=0;i<int(n);i++)
#define rep1(i,n) for(int i=1;i<=int(n);i++)
#define per(i,n) for(int i=int(n)-1;i>=0;i--)
#define per1(i,n) for(int i=int(n);i>0;i--)
#define all(c) c.begin(),c.end()
#define si(x) int(x.size())
#define pb push_back
#define eb emplace_back
#define fs first
#define sc second
template<class T> using V = vector<T>;
template<class T> using VV = vector<vector<T>>;
template<class T,class U> bool chmax(T& x, U y){
if(x<y){ x=y; return true; }
return false;
}
template<class T,class U> bool chmin(T& x, U y){
if(y<x){ x=y; return true; }
return false;
}
template<class T> void mkuni(V<T>& v){sort(all(v));v.erase(unique(all(v)),v.end());}
template<class T> int lwb(const V<T>& v, const T& a){return lower_bound(all(v),a) - v.begin();}
template<class T>
V<T> Vec(size_t a) {
return V<T>(a);
}
template<class T, class... Ts>
auto Vec(size_t a, Ts... ts) {
return V<decltype(Vec<T>(ts...))>(a, Vec<T>(ts...));
}
template<class S,class T> ostream& operator<<(ostream& o,const pair<S,T> &p){
return o<<"("<<p.fs<<","<<p.sc<<")";
}
template<class T> ostream& operator<<(ostream& o,const vector<T> &vc){
o<<"{";
for(const T& v:vc) o<<v<<",";
o<<"}";
return o;
}
constexpr ll TEN(int n) { return (n == 0) ? 1 : 10 * TEN(n-1); }
#ifdef LOCAL
#define show(x) cerr << "LINE" << __LINE__ << " : " << #x << " = " << (x) << endl
void dmpr(ostream& os){os<<endl;}
template<class T,class... Args>
void dmpr(ostream&os,const T&t,const Args&... args){
os<<t<<" ~ ";
dmpr(os,args...);
}
#define shows(...) cerr << "LINE" << __LINE__ << " : ";dmpr(cerr,##__VA_ARGS__)
#define dump(x) cerr << "LINE" << __LINE__ << " : " << #x << " = {"; \
for(auto v: x) cerr << v << ","; cerr << "}" << endl;
#else
#define show(x) void(0)
#define dump(x) void(0)
#define shows(...) void(0)
#endif
template<class D> D divFloor(D a, D b){
return a / b - (((a ^ b) < 0 && a % b != 0) ? 1 : 0);
}
template<class D> D divCeil(D a, D b) {
return a / b + (((a ^ b) > 0 && a % b != 0) ? 1 : 0);
}
template<unsigned int mod_>
struct ModInt{
using uint = unsigned int;
using ll = long long;
using ull = unsigned long long;
constexpr static uint mod = mod_;
uint v;
ModInt():v(0){}
ModInt(ll _v):v(normS(_v%mod+mod)){}
explicit operator bool() const {return v!=0;}
static uint normS(const uint &x){return (x<mod)?x:x-mod;} // [0 , 2*mod-1] -> [0 , mod-1]
static ModInt make(const uint &x){ModInt m; m.v=x; return m;}
ModInt operator+(const ModInt& b) const { return make(normS(v+b.v));}
ModInt operator-(const ModInt& b) const { return make(normS(v+mod-b.v));}
ModInt operator-() const { return make(normS(mod-v)); }
ModInt operator*(const ModInt& b) const { return make((ull)v*b.v%mod);}
ModInt operator/(const ModInt& b) const { return *this*b.inv();}
ModInt& operator+=(const ModInt& b){ return *this=*this+b;}
ModInt& operator-=(const ModInt& b){ return *this=*this-b;}
ModInt& operator*=(const ModInt& b){ return *this=*this*b;}
ModInt& operator/=(const ModInt& b){ return *this=*this/b;}
ModInt& operator++(int){ return *this=*this+1;}
ModInt& operator--(int){ return *this=*this-1;}
template<class T> friend ModInt operator+(T a, const ModInt& b){ return (ModInt(a) += b);}
template<class T> friend ModInt operator-(T a, const ModInt& b){ return (ModInt(a) -= b);}
template<class T> friend ModInt operator*(T a, const ModInt& b){ return (ModInt(a) *= b);}
template<class T> friend ModInt operator/(T a, const ModInt& b){ return (ModInt(a) /= b);}
ModInt pow(ll p) const {
if(p<0) return inv().pow(-p);
ModInt a = 1;
ModInt x = *this;
while(p){
if(p&1) a *= x;
x *= x;
p >>= 1;
}
return a;
}
ModInt inv() const { // should be prime
return pow(mod-2);
}
// ll extgcd(ll a,ll b,ll &x,ll &y) const{
// ll p[]={a,1,0},q[]={b,0,1};
// while(*q){
// ll t=*p/ *q;
// rep(i,3) swap(p[i]-=t*q[i],q[i]);
// }
// if(p[0]<0) rep(i,3) p[i]=-p[i];
// x=p[1],y=p[2];
// return p[0];
// }
// ModInt inv() const {
// ll x,y;
// extgcd(v,mod,x,y);
// return make(normS(x+mod));
// }
bool operator==(const ModInt& b) const { return v==b.v;}
bool operator!=(const ModInt& b) const { return v!=b.v;}
bool operator<(const ModInt& b) const { return v<b.v;}
friend istream& operator>>(istream &o,ModInt& x){
ll tmp;
o>>tmp;
x=ModInt(tmp);
return o;
}
friend ostream& operator<<(ostream &o,const ModInt& x){ return o<<x.v;}
};
using mint = ModInt<998244353>;
//using mint = ModInt<1000000007>;
V<mint> fact,ifact,invs;
mint Choose(int a,int b){
if(b<0 || a<b) return 0;
return fact[a] * ifact[b] * ifact[a-b];
}
void InitFact(int N){ //[0,N]
N++;
fact.resize(N);
ifact.resize(N);
invs.resize(N);
fact[0] = 1;
rep1(i,N-1) fact[i] = fact[i-1] * i;
ifact[N-1] = fact[N-1].inv();
for(int i=N-2;i>=0;i--) ifact[i] = ifact[i+1] * (i+1);
rep1(i,N-1) invs[i] = fact[i-1] * ifact[i];
}
// inplace_fmt (without bit rearranging)
// fft:
// a[rev(i)] <- \sum_j \zeta^{ij} a[j]
// invfft:
// a[i] <- (1/n) \sum_j \zeta^{-ij} a[rev(j)]
// These two are inversions.
// !!! CHANGE IF MOD is unusual !!!
const int ORDER_2_MOD_MINUS_1 = 23; // ord_2 (mod-1)
const mint PRIMITIVE_ROOT = 3; // primitive root of (Z/pZ)*
void fft(V<mint>& a){
static constexpr uint mod = mint::mod;
static constexpr uint mod2 = mod + mod;
static const int H = ORDER_2_MOD_MINUS_1;
static const mint root = PRIMITIVE_ROOT;
static mint magic[H-1];
int n = si(a);
assert(!(n & (n-1))); assert(n >= 1); assert(n <= 1<<H); // n should be power of 2
if(!magic[0]){ // precalc
rep(i,H-1){
mint w = -root.pow(((mod-1)>>(i+2))*3);
magic[i] = w;
}
}
int m = n;
if(m >>= 1){
rep(i,m){
uint v = a[i+m].v; // < M
a[i+m].v = a[i].v + mod - v; // < 2M
a[i].v += v; // < 2M
}
}
if(m >>= 1){
mint p = 1;
for(int h=0,s=0; s<n; s += m*2){
for(int i=s;i<s+m;i++){
uint v = (a[i+m] * p).v; // < M
a[i+m].v = a[i].v + mod - v; // < 3M
a[i].v += v; // < 3M
}
p *= magic[__builtin_ctz(++h)];
}
}
while(m){
if(m >>= 1){
mint p = 1;
for(int h=0,s=0; s<n; s += m*2){
for(int i=s;i<s+m;i++){
uint v = (a[i+m] * p).v; // < M
a[i+m].v = a[i].v + mod - v; // < 4M
a[i].v += v; // < 4M
}
p *= magic[__builtin_ctz(++h)];
}
}
if(m >>= 1){
mint p = 1;
for(int h=0,s=0; s<n; s += m*2){
for(int i=s;i<s+m;i++){
uint v = (a[i+m] * p).v; // < M
a[i].v = (a[i].v >= mod2) ? a[i].v - mod2 : a[i].v; // < 2M
a[i+m].v = a[i].v + mod - v; // < 3M
a[i].v += v; // < 3M
}
p *= magic[__builtin_ctz(++h)];
}
}
}
rep(i,n){
a[i].v = (a[i].v >= mod2) ? a[i].v - mod2 : a[i].v; // < 2M
a[i].v = (a[i].v >= mod) ? a[i].v - mod : a[i].v; // < M
}
// finally < mod !!
}
void invfft(V<mint>& a){
static constexpr uint mod = mint::mod;
static constexpr uint mod2 = mod + mod;
static const int H = ORDER_2_MOD_MINUS_1;
static const mint root = PRIMITIVE_ROOT;
static mint magic[H-1];
int n = si(a);
assert(!(n & (n-1))); assert(n >= 1); assert(n <= 1<<H); // n should be power of 2
if(!magic[0]){ // precalc
rep(i,H-1){
mint w = -root.pow(((mod-1)>>(i+2))*3);
magic[i] = w.inv();
}
}
int m = 1;
if(m < n>>1){
mint p = 1;
for(int h=0,s=0; s<n; s += m*2){
for(int i=s;i<s+m;i++){
ull x = a[i].v + mod - a[i+m].v; // < 2M
a[i].v += a[i+m].v; // < 2M
a[i+m].v = (p.v * x) % mod; // < M
}
p *= magic[__builtin_ctz(++h)];
}
m <<= 1;
}
for(;m < n>>1; m <<= 1){
mint p = 1;
for(int h=0,s=0; s<n; s+= m*2){
for(int i=s;i<s+(m>>1);i++){
ull x = a[i].v + mod2 - a[i+m].v; // < 4M
a[i].v += a[i+m].v; // < 4M
a[i].v = (a[i].v >= mod2) ? a[i].v - mod2 : a[i].v; // < 2M
a[i+m].v = (p.v * x) % mod; // < M
}
for(int i=s+(m>>1); i<s+m; i++){
ull x = a[i].v + mod - a[i+m].v; // < 2M
a[i].v += a[i+m].v; // < 2M
a[i+m].v = (p.v * x) % mod; // < M
}
p *= magic[__builtin_ctz(++h)];
}
}
if(m < n){
rep(i,m){
uint x = a[i].v + mod2 - a[i+m].v; // < 4M
a[i].v += a[i+m].v; // < 4M
a[i+m].v = x; // < 4M
}
}
const mint in = mint(n).inv();
rep(i,n) a[i] *= in; // < M
// finally < mod !!
}
// A,B = 500000 -> 70ms
// verify https://judge.yosupo.jp/submission/44937
V<mint> multiply(V<mint> a, V<mint> b) {
int A = si(a), B = si(b);
if (!A || !B) return {};
int n = A+B-1;
int s = 1; while(s<n) s*=2;
if(a == b){ // # of fft call : 3 -> 2
a.resize(s); fft(a);
rep(i,s) a[i] *= a[i];
}else{
a.resize(s); fft(a);
b.resize(s); fft(b);
rep(i,s) a[i] *= b[i];
}
invfft(a); a.resize(n);
return a;
}
/*
係数アクセス
f[i] でいいが、 配列外参照する可能性があるなら at/set
*/
template<class mint>
struct Poly: public V<mint>{
using vector<mint>::vector;
Poly() {}
explicit Poly(int n) : V<mint>(n){} // poly<mint> a; a = 2; shouldn't be [0,0]
Poly(int n, mint c) : V<mint>(n,c){}
Poly(const V<mint>& a) : V<mint>(a){}
Poly(initializer_list<mint> li) : V<mint>(li){}
int size() const { return V<mint>::size(); }
mint at(int i) const {
return i<size() ? (*this)[i] : 0;
}
void set(int i, mint x){
if(i>=size() && !x) return;
while(i>=size()) this->pb(0);
(*this)[i] = x;
return;
}
mint operator()(mint x) const { // eval
mint res = 0;
int n = size();
mint a = 1;
rep(i,n){
res += a * (*this)[i];
a *= x;
}
return res;
}
Poly low(int n) const { // ignore x^n (take first n), but not empty
return Poly(this->begin(), this->begin()+min(max(n,1),size()));
}
Poly rev() const {
return Poly(this->rbegin(), this->rend());
}
friend ostream& operator<<(ostream &o,const Poly& f){
o << "[";
rep(i,f.size()){
o << f[i];
if(i != f.size()-1) o << ",";
}
o << "]";
return o;
}
Poly operator-() const {
Poly res = *this;
for(auto& v: res) v = -v;
return res;
}
Poly& operator+=(const mint& c){
(*this)[0] += c;
return *this;
}
Poly& operator-=(const mint& c){
(*this)[0] -= c;
return *this;
}
Poly& operator*=(const mint& c){
for(auto& v: *this) v *= c;
return *this;
}
Poly& operator/=(const mint& c){
return *this *= mint(1)/mint(c);
}
Poly& operator+=(const Poly& r){
if(size() < r.size()) this->resize(r.size(),0);
rep(i,r.size()) (*this)[i] += r[i];
return *this;
}
Poly& operator-=(const Poly& r){
if(size() < r.size()) this->resize(r.size(),0);
rep(i,r.size()) (*this)[i] -= r[i];
return *this;
}
Poly& operator*=(const Poly& r){
return *this = multiply(*this,r);
}
// 何回も同じrで割り算するなら毎回rinvを計算するのは無駄なので、呼び出し側で一回計算した後直接こっちを呼ぶと良い
// 取るべきinvの長さに注意
// 例えば mod r で色々計算したい時は、基本的に deg(r) * 2 長さの多項式を r で割ることになる
// とはいえいったん rinv を長く計算したらより短い場合はprefix見るだけだし、 rinv としてムダに長いものを渡しても問題ないので
// 割られる多項式として最大の次数を取ればよい
Poly quotient(const Poly& r, const Poly& rinv){
int m = r.size(); assert(r[m-1].v);
int n = size();
int s = n-m+1;
if(s <= 0) return {0};
return (rev().low(s)*rinv.low(s)).low(s).rev();
}
Poly& operator/=(const Poly& r){
return *this = quotient(r,r.rev().inv(max(size()-r.size(),0)+1));
}
Poly& operator%=(const Poly& r){
*this -= *this/r * r;
return *this = low(r.size()-1);
}
Poly operator+(const mint& c) const {return Poly(*this) += c; }
Poly operator-(const mint& c) const {return Poly(*this) -= c; }
Poly operator*(const mint& c) const {return Poly(*this) *= c; }
Poly operator/(const mint& c) const {return Poly(*this) /= c; }
Poly operator+(const Poly& r) const {return Poly(*this) += r; }
Poly operator-(const Poly& r) const {return Poly(*this) -= r; }
Poly operator*(const Poly& r) const {return Poly(*this) *= r; }
Poly operator/(const Poly& r) const {return Poly(*this) /= r; }
Poly operator%(const Poly& r) const {return Poly(*this) %= r; }
Poly diff() const {
Poly g(max(size()-1,0));
rep(i,g.size()) g[i] = (*this)[i+1] * (i+1);
return g;
}
Poly intg() const {
assert(si(invs) > size());
Poly g(size()+1);
rep(i,size()) g[i+1] = (*this)[i] * invs[i+1];
return g;
}
Poly square() const {
return multiply(*this,*this);
}
// 1/f(x) mod x^s
// N = s = 500000 -> 90ms
// inv は 5 回 fft(2n) を呼んでいるので、multiply が 3 回 fft(2n) を呼ぶのと比べると
// だいたい multiply の 5/3 倍の時間がかかる
// 導出: Newton
// fg = 1 mod x^m
// (fg-1)^2 = 0 mod x^2m
// f(2g-fg^2) = 1 mod x^2m
// verify: https://judge.yosupo.jp/submission/44938
Poly inv(int s) const {
Poly r(s);
r[0] = mint(1)/at(0);
for(int n=1;n<s;n*=2){ // 5 times fft : length 2n
V<mint> f = low(2*n); f.resize(2*n);
fft(f);
V<mint> g = r.low(2*n); g.resize(2*n);
fft(g);
rep(i,2*n) f[i] *= g[i];
invfft(f);
rep(i,n) f[i] = 0;
fft(f);
rep(i,2*n) f[i] *= g[i];
invfft(f);
for(int i=n;i<min(2*n,s);i++) r[i] -= f[i];
}
return r;
}
// log f mod x^s
// 導出: D log(f) = (D f) / f
// 500000: 180ms
// mult の 8/3 倍
// verify: https://judge.yosupo.jp/submission/44962
Poly log(int s) const {
assert(at(0) == 1);
if(s == 1) return {0};
return (low(s).diff() * inv(s-1)).low(s-1).intg();
}
// e^f mod x^s
// f.log(s).exp(s) == [1,0,...,0]
// 500000 : 440ms
// TODO: 高速化！
// 速い実装例 (hos): https://judge.yosupo.jp/submission/36732 150ms
// 導出 Newton:
// g = exp(f)
// log(g) - f = 0
// g == g0 mod x^m
// g == g0 - (log(g0) - f) / (1/g0) mod x^2m
// verify: yosupo
Poly exp(int s) const {
assert(at(0) == 0);
Poly f({1}),g({1});
for(int n=1;n<s;n*=2){
g = (g*2-g.square().low(n)*f).low(n);
Poly q = low(n).diff();
q = q + g * (f.diff() - f*q).low(2*n-1);
f = (f + f * (low(2*n)-q.intg()) ).low(2*n);
}
return f.low(s);
}
// f^p mod x^s
// 500000: 600ms
// 導出: f^p = e^(p log f)
// log 1回、 exp 1回
// Exp.cpp (Mifafa technique) も参照
// c.f. (f の non0 coef の個数) * s
// verify: https://judge.yosupo.jp/submission/44992
Poly pow(ll p, int s) const {
if(p == 0){
return Poly(s) + 1; // 0^0 is 1
}
int ord = 0;
while(ord<s && !at(ord)) ord++;
if((s-1)/p < ord) return Poly(s); // s <= p * ord
int off = p*ord;
int s_ = s-off;
const mint a0 = at(ord), ia0 = a0.inv(), ap = a0.pow(p);
Poly f(s_); rep(i,s_) f[i] = at(i+ord) * ia0;
f = (f.log(s_) * p).exp(s_);
Poly res(s);
rep(i,s_) res[i+off] = f[i] * ap;
return res;
}
// f^(1/2) mod x^s
// f[0] should be 1
// 11/6
// verify: https://judge.yosupo.jp/submission/44997
Poly sqrt(int s) const {
assert(at(0) == 1);
static const mint i2 = mint(2).inv();
V<mint> f{1},g{1},z{1};
for(int n=1;n<s;n*=2){
rep(i,n) z[i] *= z[i];
invfft(z);
V<mint> d(2*n);
rep(i,n) d[n+i] = z[i] - at(i) - at(n+i);
fft(d);
V<mint> g2(2*n);
rep(i,n) g2[i] = g[i];
fft(g2);
rep(i,n*2) d[i] *= g2[i];
invfft(d);
f.resize(n*2);
for(int i=n;i<n*2;i++) f[i] = -d[i] * i2;
if(n*2 >= s) break;
z = f;
fft(z);
V<mint> eps = g2;
rep(i,n*2) eps[i] *= z[i];
invfft(eps);
rep(i,n) eps[i] = 0;
fft(eps);
rep(i,n*2) eps[i] *= g2[i];
invfft(eps);
g.resize(n*2);
for(int i=n;i<n*2;i++) g[i] -= eps[i];
}
f.resize(s);
return f;
}
// Taylor Shift
// return f(x+c)
// O(N logN)
// verify: yosupo
Poly shift(mint c){
int n = size();
assert(si(fact) >= n); // please InitFact
V<mint> f(n); rep(i,n) f[i] = (*this)[i] * fact[i];
V<mint> g(n);
mint cpow = 1;
rep(i,n){g[i] = cpow * ifact[i]; cpow *= c;}
reverse(all(g));
V<mint> h = multiply(f,g);
Poly res(n); rep(i,n) res[i] = h[n-1+i] * ifact[i];
return res;
}
};
// return f(K+1)
// f[k] = 0^k + .. + n^k
// \sum_{k>=0} f[k] x^k/k! = e^0x + .. + e^nx = 1-e^(n+1)x / 1-e^x
// O(KlogK)
// 0^0 = 1
// keyword: faulhaber ファウルハーバー
vector<mint> SumOfPower(mint n, int K){
assert(si(fact) > K);
Poly<mint> a(K+1),b(K+1);
mint pw = 1;
rep1(i,K+1){
pw *= n+1;
a[i-1] = ifact[i];
b[i-1] = ifact[i] * pw;
}
auto f = b*a.inv(K+1);
V<mint> res(K+1);
rep(k,K+1) res[k] = f[k] * fact[k];
return res;
}
int main(){
cin.tie(0);
ios::sync_with_stdio(false); //DON'T USE scanf/printf/puts !!
cout << fixed << setprecision(20);
InitFact(300000);
int N; ll M; cin >> N >> M;
queue<Poly<mint>> q; q.push(Poly<mint>({1}));
rep(i,N-1){
q.push(Poly<mint>({M+1+i,-1}));
}
while(si(q) > 1){
auto f = q.front(); q.pop();
auto g = q.front(); q.pop();
q.push(f*g);
}
auto f = q.front();
f *= ifact[N-1];
Poly<mint> g(5001);
rep(i,N){
int k; cin >> k; g[k]++;
}
f *= g;
int X = si(f)-1;
auto p = SumOfPower(M,X);
mint ans = 0;
rep(i,X+1) ans += f[i] * p[i];
cout << ans << endl;
}