#1193408 (C++23) No.3767 Earth Mover's Distance Sum

• 問題: No.3767 Earth Mover's Distance Sum
• ユーザー: aPNJ777
• 提出日時: 2026-10-17 00:52:11
• 言語: C++23 (gcc 15.3.0 + boost 1.92.0 + ACL)
• 結果: AC
• 実行時間: 215 ms / 2,000 ms
• コード長: 15,877 bytes
• コンパイル時間: 2,980 ms
• コンパイル使用メモリ: 346,340 KB
• 実行使用メモリ: 30,748 KB
• 最終ジャッジ日時: 2026-10-17 00:52:25
• 合計ジャッジ時間: 10,271 ms
• ジャッジサーバーID （参考情報）: judge4_0 / judge2_0

ファイルパターン	結果
sample	AC * 3
other	AC * 29

ソースコード

//////////////////////////////////////// template start /////////////////////////////////////////
// https://atcoder.jp/contests/abc479/submissions/80038224 2026-10-14 19:59:22
#include <bits/stdc++.h>
using namespace std;

//using ll = long long;
using ld = long double;
using u32 = uint32_t;
using u64 = uint64_t;
using i128 = __int128_t;
using u128 = __uint128_t;

template <class T>
using vc = vector<T>;
template <class T>
using vvc = vector<vc<T>>;
template <class T>
using vvvc = vector<vvc<T>>;
template <class T>
using vvvvc = vector<vvvc<T>>;
template <class T>
using vvvvvc = vector<vvvvc<T>>;

#define vv(type, name, h, w) vector<vector<type>> name(h, vector<type>(w))
#define vvv(type, name, h, w, l) vector<vector<vector<type>>> name(h, vector<vector<type>>(w, vector<type>(l)))
#define vvvv(type, name, a, b, c, d) vector<vector<vector<vector<type>>>> name(a, vector<vector<vector<type>>>(b, vector<vector<type>>(c, vector<type>(d))))
#define vvvvv(type, name, a, b, c, d, e) vector<vector<vector<vector<vector<type>>>>> name(a, vector<vector<vector<vector<type>>>>(b, vector<vector<vector<type>>>(c, vector<vector<type>>(d, vector<type>(e)))))

#define elif else if

#define FOR1(a) for (long long _ = 0; _ < (long long)(a); _++)
#define FOR2(i, n) for (long long i = 0; i < (long long)(n); i++)
#define FOR3(i, l, r) for (long long i = l; i < (long long)(r); i++)
#define FOR4(i, l, r, c) for (long long i = l; i < (long long)(r); i += c)
#define FOR1_R(a) for (long long _ = (long long)(a) - 1; _ >= 0; _--)
#define FOR2_R(i, n) for (long long i = (long long)(n) - 1; i >= (long long)(0); i--)
#define FOR3_R(i, l, r) for (long long i = (long long)(r) - 1; i >= (long long)(l); i--)
#define FOR4_R(i, l, r, c) for (long long i = (long long)(r) - 1; i >= (long long)(l); i -= (c))
#define overload4(a, b, c, d, e, ...) e
#define FOR(...) overload4(__VA_ARGS__, FOR4, FOR3, FOR2, FOR1)(__VA_ARGS__)
#define FOR_R(...) overload4(__VA_ARGS__, FOR4_R, FOR3_R, FOR2_R, FOR1_R)(__VA_ARGS__)
#define FOR_in(a, A) for (auto a: A)
#define FOR_each(a, A) for (auto &&a: A)
#define FOR_subset(t, s) for(long long t = (s); t >= 0; t = (t == 0 ? -1 : (t - 1) & (s)))

#define all(x) x.begin(), x.end()
#define len(x) int(x.size())

int popcount(int x) { return __builtin_popcount(x); }
int popcount(uint32_t x) { return __builtin_popcount(x); }
int popcount(long long x) { return __builtin_popcountll(x); }
int popcount(uint64_t x) { return __builtin_popcountll(x); }
// __builtin_clz(x)は最上位bitからいくつ0があるか.
int topbit(int x) { return (x == 0 ? -1 : 31 - __builtin_clz(x)); }
int topbit(uint32_t x) { return (x == 0 ? -1 : 31 - __builtin_clz(x)); }
int topbit(long long x) { return (x == 0 ? -1 : 63 - __builtin_clzll(x)); }
int topbit(uint64_t x) { return (x == 0 ? -1 : 63 - __builtin_clzll(x)); }

// 入力
void rd() {}
void rd(char &c) { cin >> c; }
void rd(string &s) { cin >> s; }
void rd(int &x) { cin >> x; }
void rd(uint32_t &x) { cin >> x; }
void rd(long long &x) { cin >> x; }
void rd(uint64_t &x) { cin >> x; }
template<class T>
void rd(vector<T> &v) { for (auto& x:v) rd(x); }

void read() {}
template <class H, class... T>
void read(H &h, T &... t) { rd(h), read(t...); }

#define CHAR(...) \
  char __VA_ARGS__; \
  read(__VA_ARGS__)

#define STRING(...) \
  string __VA_ARGS__; \
  read(__VA_ARGS__)

#define INT(...) \
  int __VA_ARGS__; \
  read(__VA_ARGS__)

#define U32(...) \
  uint32_t __VA_ARGS__; \
  read(__VA_ARGS__)

#define LL(...) \
  long long __VA_ARGS__; \
  read(__VA_ARGS__)

#define U64(...) \
  uint64_t __VA_ARGS__; \
  read(__VA_ARGS__)

#define VC(t, a, n) \
  vector<t> a(n); \
  read(a)

#define VVC(t, a, h, w) \
  vector<vector<t>> a(h, vector<t>(w)); \
  read(a)

//出力
void wt() {}
void wt(const char c) { cout << c; }
void wt(const string s) { cout << s; }
void wt(int x) { cout << x; }
void wt(uint32_t x) { cout << x; }
void wt(long long x) { cout << x; }
void wt(uint64_t x) { cout << x; }
void wt(double x) { cout << fixed << setprecision(16) << x; }
void wt(long double x) { cout << fixed << setprecision(16) << x; }

template<class T>
void wt(const vector<T> v) {
  int n = v.size();
  for (int i = 0; i < n; i++) {
    if (i) wt(' ');
    wt(v[i]);
  }
}

void print() { wt('\n'); }
template <class Head, class... Tail>
void print(Head &&head, Tail &&... tail) {
  wt(head);
  if (sizeof...(Tail)) wt(' ');
  print(forward<Tail>(tail)...);
}

template <class T>
int bisect_left(const vector<T> & A, T x) {
  auto idx = lower_bound(A.begin(), A.end(), x) - A.begin();
  return idx;
}

template <class T>
int bisect_right(const vector<T> & A, T x) {
  auto idx = upper_bound(A.begin(), A.end(), x) - A.begin();
  return idx;
}

template <class T>
T min(vector<T> A) {
  assert (A.size());
  T S = A[0];
  for (T a : A) S = min(a, S);
  return S;
}

template <class T>
T max(vector<T> A) {
  assert (A.size());
  T S = A[0];
  for (T a : A) S = max(a, S);
  return S;
}

template <class T>
T add(T x, T y) { return x + y; }

template <class T>
bool chmin(T & x, T a) { return a < x ? (x = a, true) : false; }

template <class T>
bool chmax(T & x, T a) { return a > x ? (x = a, true) : false; }

template <class T>
T sum(vector<T> A) {
  T S = 0;
  for (int i = 0; i < int(A.size()); i++) S += A[i];
  return S;
}

uint64_t random_u64(uint64_t l, uint64_t r) {
  static std::random_device rd;
  static std::mt19937_64 gen(rd());
  std::uniform_int_distribution<uint64_t> dist(l, r);
  return dist(gen);
}

long long floor_div(long long a, long long b) {
  if (b < 0) a *= -1, b *= -1;
  if (a >= 0) return a / b;
  return -((-a + b - 1) / b);
}

long long ceil_div(long long a, long long b) {
    if (b < 0) a *= -1, b *= -1;
    if (a >= 0) return (a + b - 1) / b;
    return - ((-a) / b);
}

long long gcd(long long a, long long b) {
  while (a) {
    b %= a;
    if (b == 0) return a;
    a %= b;
  }
  return b;
}

long long lcm(long long a, long long b) {
  if (a * b == 0) return 0;
  return a * b / gcd(a, b);
}

long long pow_mod(long long a, long long r, long long mod) {
  long long res = 1, p = a % mod;
  while (r) {
    if ((r % 2) == 1) res = res * p % mod;
    p = p * p % mod, r >>= 1;
  }
  return res;
}

long long mod_inv(long long a, long long mod) {
  if (mod == 1) return 0;
  a %= mod;
  long long b = mod, s = 1, t = 0;
  while (1) {
    if (a == 1) return s;
    t -= (b / a) * s;
    b %= a;
    if (b == 1) return t + mod;
    s -= (a / b) * t;
    a %= b;
  }
}

pair<long long, long long> extgcd(long long a, long long b) {
  long long s = a, t = b;
  long long xs = 1, ys = 0, xt = 0, yt = 1;
  while (t > 0) {
    long long u = s / t;
    s -= t * u;
    xs -= xt * u;
    ys -= yt * u;
    swap(s, t);
    swap(xs, xt), swap(ys, yt);
  }
  return {xs, ys};
}

pair<long long, long long> garner(vector<long long> Rem, vector<long long> Mod) {
  assert (Rem.size() == Mod.size());
  int n = int(Mod.size());
  // garner
  long long x = 0, prod = 1;
  for (int i = 0; i < n; i++) {
    if (Mod[i] == 1) continue;
    long long g = gcd(prod, Mod[i]);
    long long c = (Rem[i] + (Mod[i] - x % Mod[i])) % Mod[i];
    if (c % g) return {-1, -1}; // not exist
    long long m = Mod[i] / g;
    c = (c / g) * mod_inv((prod / g) % m, m) % m;
    x = x + c * prod;
    prod *= m;
    x %= prod;
  }
  return {x, prod};
}

long long garner2(vector<long long> Rem, vector<long long> Mod, long long mod) {
  assert (Rem.size() == Mod.size());
  int n = int(Mod.size());
  // garner
  vector<long long> C(n);
  for (int i = 0; i < n; i++) {
    long long x = 0, prod = 1;
    for (int j = 0; j < i; j++) {
      x = (x + C[j] * prod % Mod[i]) % Mod[i];
      prod = (prod * Mod[j]) % Mod[i];
    }
    long long g = gcd(prod, Mod[i]);
    long long c = (Rem[i] + (Mod[i] - x)) % Mod[i];
    if (c % g) return -1; // not exist
    long long m = Mod[i] / g;
    c = (c / g) * mod_inv((prod / g) % m, m) % m;
    Mod[i] /= g, C[i] = c;
  }
  long long ans = 0, c = 0;
  for (int i = n - 1; i >= 0; i--) ans = ((ans * Mod[i]) % mod + C[i]) % mod, c = max(c, C[i]);
  // 中華風
  if (c == 0) {
    ans = 1;
    for (int i = 0; i < n; i++) ans = (ans * Mod[i]) % mod;
  }
  return ans;
}

long long Tonelli_Shanks(long long a, long long mod) {
  a %= mod;
  if (a < 2) return a;
  if (pow_mod(a, (mod - 1) / 2, mod) != 1) return -1;
  if (mod % 4 == 3) return pow_mod(a, (mod + 1) / 4, mod);

  long long b = 3;
  if (mod != 998244353) {
    while (pow_mod(b, (mod - 1) / 2, mod) == 1) b = random_u64(2, mod - 1);
  }

  long long q = mod - 1;
  long long Q = 0;
  while (q % 2 == 0) Q++, q /= 2;

  long long x = pow_mod(a, (q + 1) / 2, mod);
  b = pow_mod(b, q, mod);

  long long shift = 2;
  while ((x * x) % mod != a) {
    long long error = (((pow_mod(a, mod - 2, mod) * x) % mod) * x) % mod;
    if (pow_mod(error, 1 << (Q - shift), mod) != 1) x = (x * b) % mod;
    b = (b * b) % mod;
    shift++;
  }
  return x;
}

//////////////////////////////////////// template end ////////////////////////////////////////

//////////////////////////////////////// modint start ////////////////////////////////////////

template <int mod>
struct modint {
  static constexpr uint32_t umod = uint32_t(mod);
  static_assert(umod < (uint32_t(1) << 31));
  uint32_t val;

  static modint raw(uint32_t v) {
    modint x;
    x.val = v % umod;
    return x;
  }

  constexpr modint() : val(0) {}
  constexpr modint(uint32_t x) : val(x % umod) {}
  constexpr modint(uint64_t x) : val(x % umod) {}
  constexpr modint(__uint128_t x) : val(x % umod) {}
  constexpr modint(int x) : val((x %= int(umod)) < 0 ? x + umod : x) {};
  constexpr modint(long long x) : val((x %= int(umod)) < 0 ? x + umod : x) {};
  constexpr modint(__int128_t x) : val((x %= int(umod)) < 0 ? x + umod : x) {};

  bool operator<(const modint &other) const { return val < other.val; }
  modint &operator+=(const modint &p) {
    if ((val += p.val) >= umod) val -= umod;
    return *this;
  }
  modint &operator-=(const modint &p) {
    if ((val += umod - p.val) >= umod) val -= umod;
    return *this;
  }
  modint &operator*=(const modint &p) {
    val = uint64_t(val) * p.val % umod;
    return *this;
  }
  modint &operator/=(const modint &p) {
    val = uint64_t(val) * p.inverse().val % umod;
    return *this;
  }
  modint operator-() const { return modint::raw(val ? umod - val : uint32_t(0)); }
  modint operator+(const modint &p) const { return modint(*this) += p; }
  modint operator-(const modint &p) const { return modint(*this) -= p; }
  modint operator*(const modint &p) const { return modint(*this) *= p; }
  modint operator/(const modint &p) const { return modint(*this) /= p; }
  bool operator==(const modint &p) const { return val == p.val; }
  bool operator!=(const modint &p) const { return val != p.val; }

  modint inverse() const {
    int a = val, b = umod, s = 1, t = 0;
    while (1) {
      if (a == 1) return modint(s);
      t -= (b / a) * s;
      b %= a;
      if (b == 1) return modint(t + umod);
      s -= (a / b) * t;
      a %= b;
    }
  }

  modint pow(long long n) const {
    if (n == 0) return modint(1);
    if (val == 0) return modint(0);
    n %= (long long)(umod - 1);
    if (n < 0) n += umod - 1;
    modint res(1), a(val);
    while (n > 0) {
      if (n & 1) res *= a;
      a *= a;
      n >>= 1;
    }
    return res;
  }

  uint32_t get() const { return val; }

  static constexpr int get_mod() { return mod; }
  
  static constexpr pair<int, int> ntt_info() {
    if (mod == 167772161) return {25, 17};
    if (mod == 469762049) return {26, 30};
    if (mod == 754974721) return {24, 362};
    if (mod == 880803841) return {23, 211};
    if (mod == 998244353) return {23, 31};
    return {-1, -1};
  }
};

template <int mod>
void rd(modint<mod> &x) {
  uint32_t y;
  cin >> y;
  x = y;
}

template <int mod>
void wt(modint<mod> x) { wt(x.val); }

template <class mint>
mint inverse(long long n) {
  static bool prepared = 0;
  static int N = 3000000, mod = mint::get_mod();
  if (N > mod - 1) N = mod - 1;
  static vector<mint> res(N + 1, mint(1));
  if (prepared == 0) {
    prepared = 1;
    for (int i = 2; i <= N; i++) res[i] = -res[mod % i] * mint(mod / i);
  }
  if (n < 0) return mint(0);
  if (n == 0) return mint(1); /////////////////
  assert (n <= N);
  return res[n];
}

template <typename mint>
mint fact(long long n) {
  static bool prepared = 0;
  static int N = 3000000, mod = mint::get_mod();
  if (N > mod - 1) N = mod - 1;
  static vector<mint> res(N + 1, mint(1));
  if (prepared == 0) {
    prepared = 1;
    for (int i = 2; i <= N; i++) res[i] = res[i - 1] * mint(i);
  }
  if (n < 0) return mint(0);
  assert (n <= N);
  return res[n];
}

template <typename mint>
mint fact_inv(long long n) {
  static bool prepared = 0;
  static int N = 3000000, mod = mint::get_mod();
  if (N > mod - 1) N = mod - 1;
  static vector<mint> res(N + 1, mint(1));
  if (prepared == 0) {
    prepared = 1;
    res[N] = fact<mint>(N).inverse();
    for (int i = N - 1; i >= 1; i--) res[i] = res[i + 1] * mint(i + 1);
  }
  if (n < 0) return mint(0);
  assert (n <= N);
  return res[n];
}

template <typename mint>
mint binom(long long n, long long r) {
  if (n < 0 || r < 0 || n < r) return 0;
  mint res = fact<mint>(n) * (fact_inv<mint>(n - r) * fact_inv<mint>(r));
  return res;
}

//////////////////////////////////////// modint end ////////////////////////////////////////

using mint = modint<998244353>;

/*
検索するとき

https://www.google.com/search?udm=14&q=

-ai

難読化???

todo ABCまでに github に code
*/

/*
貪欲にやればいい.

B_1 を決める.
A_1 が B_1 に (1, 2) で操作
A は B_1, A_1 + A_2 - B_1, ...
A_2 を B_2 に
B_1, B_2, A_1 + A_2 + A_3 - (B_1 + B_2)

操作回数は \sum (|A_1 + ... + A_i - (B_1 + ... + B_i)|)

|(A_1 + ... + A_i) - (B_1 + ... + B_i)| の寄与を求めればいい.

i 毎 a_i = A_1 + ... + A_i

a の寄与 B_i までの和 a以下のとき-, a以上+

fps
1/(1-x)^i の [x^a] まで と 1/(1-x)^{N-i} の積の [x^S]
i が 1 増える 左辺の変化, -binom(i+a,a) x^{a+1} を引いて, (1-x)で割る.
a が増える 左辺の変化 binom(i - 1 + a, i)x^a 足す.

ix /(1-x)^{i+1} の [x^a] まで と 1/(1-x)^{N-i} の積の [x^S]
i が増える. 左辺の変化, -ibinom(i-1+a,a) x^{a+1} を引いて, x /(1-x)^i の [x^a] まで足して, (1-x)で割る
*/

void solve() {
  INT(N);
  VC(int, A, N);
  for (int i = 0; i < N - 1; i++) A[i + 1] += A[i];
  int S = A.back();
  mint ans = mint(0);
  // b - a としたときの答え
  for (int i = 1; i < N; i++) {
    int a = A[i - 1];
    mint res = mint(i) * binom<mint>(N + S - 1, S - 1) - mint(a) * binom<mint>(N - 1 + S, S);
    ans += res;
  }
  // + a の寄与
  int b = 0;
  mint res = binom<mint>(N - 1 + S, S);
  for (int i = 1; i < N; i++) {
    int a = A[i - 1];
    // i が増える.
    res -= binom<mint>(i - 1 + b, b) * binom<mint>(N - i + (S - (b + 1)), S - (b + 1));
    // b 増える
    while (b < a) {
      b++;
      res += binom<mint>(i - 1 + b, b) * binom<mint>(N - i - 1 + S - b , S - b);
    }
    ans += mint(2 * a) * res; 
  }
  // b の余分な寄与
  b = 1, res = binom<mint>(N - 1 + S - 1, S - 1);
  mint res1 = mint(0);
  for (int i = 1; i < N; i++) {
    int a = A[i - 1];
    res1 -= mint(i - 1) * binom<mint>(i + b - 1, b - 1) * binom<mint>(N - i + (S - (b + 1)), S - (b + 1));
    res -= binom<mint>(i + b - 1, b - 1) * binom<mint>(N - i + (S - (b + 1)), S - (b + 1));
    res1 += res;
    while (b < a) {
      b++;
      res += binom<mint>(i + b - 1, b - 1) * binom<mint>(N - i - 1 + S - b , S - b);
      res1 += mint(i) * binom<mint>(i + b - 1, b - 1) * binom<mint>(N - i - 1 + S - b , S - b);
    }
    if (a >= 1) ans -= mint(2) * res1;
  }
  print(ans);
}

int main() {
  ios::sync_with_stdio(false);
  cin.tie(nullptr);
  int testcases = 1;
  //cin >> testcases;
  FOR(testcases) solve();
}