多項式の Prefix Sum
(fps/prefix-sum-of-polynomial.hpp)

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• Last update: 2025-10-25 18:30:13+09:00
• Include: #include "fps/prefix-sum-of-polynomial.hpp"

多項式 $f$ について $g(n)=\sum_{i=0}^{n}f(i)$ を満たす多項式 $g$ を求める．

$d=\deg f$ として $O(d\log d)$ 時間．

アルゴリズム

Bernoulli 数 $B_i=\left[\frac{x^i}{i!}\right]\dfrac{xe^x}{e^x-1}$ を用いると以下が成り立つ．

\[\begin{align*} \sum_{i=1}^{n}i^k &=k![x^k]\sum_{i=1}^{n}e^{ix}\\ &=k![x^k]\frac{(e^{nx}-1)e^x}{e^x-1}\\ &=k![x^k]\frac{e^{nx}-1}{x}\cdot\frac{xe^x}{e^x-1}\\ &=k!\sum_{i=0}^{k}\frac{n^{i+1}}{(i+1)!}\cdot\frac{B_{k-i}}{(k-i)!} \end{align*}\]

これは Faulhaber の公式として知られている．

$f(x)=\sum_{i=0}^{d-1}f_ix^i$ とすると，上の結果から $1\leq l\leq d$ に対して

\[\begin{align*} [x^l]g(x) &=\frac{1}{l!}\sum_{k=l-1}^{d-1}f_kk!\cdot\frac{B_{k-(l-1)}}{(k-(l-1))!}\\ &=\frac{1}{l!}[x^{d-l}]\left(\sum_{i=0}^{d-1}f_ii!\cdot x^{d-1-i}\right)\frac{xe^x}{e^x-1} \end{align*}\]

が成り立つ．また $[x^0]g(x)=f_0$ であるから，結局 $g$ は $O(d\log d)$ 時間で計算できる．

• https://codeforces.com/blog/entry/98563
• https://www.codechef.com/problems/SERSUM

Depends on

• fps/formal-power-series.hpp
• 階乗, 二項係数 (modint/factorial.hpp)

Verified with

• verify/fps/UNIT_prefix_sum_of_polynomial.test.cpp

Code

#pragma once
#include "modint/factorial.hpp"
#include "fps/formal-power-series.hpp"

// g(n)=sum_{i=0}^{n}f(i)
template <class mint>
FormalPowerSeries<mint> PrefixSumOfPolynomial(FormalPowerSeries<mint> f) {
  if (f.empty()) return {};
  using fact = Factorial<mint>;
  mint c = f[0];
  int d = f.size();
  fact::reserve(d);
  for (int i = 0; i < d; i++) f[i] *= fact::fact(i);
  reverse(f.begin(), f.end());
  FormalPowerSeries<mint> g(d);
  for (int i = 0; i < d; i++) g[i] = fact::fact_inv(i + 1) * (i & 1 ? -1 : 1);
  f *= g.inv();
  f.resize(d);
  f.push_back(c);
  reverse(f.begin(), f.end());
  for (int i = 1; i <= d; i++) f[i] *= fact::fact_inv(i);
  return f;
}
/**
 * @brief 多項式の Prefix Sum
 * @docs docs/fps/prefix-sum-of-polynomial.md
 */

#line 2 "modint/factorial.hpp"

template <class mint>
struct Factorial {
  static void reserve(int n) {
    inv(n);
    fact(n);
    fact_inv(n);
  }
  static mint inv(int n) {
    static long long mod = mint::get_mod();
    static vector<mint> buf({0, 1});
    assert(n != 0);
    if (mod != mint::get_mod()) {
      mod = mint::get_mod();
      buf = vector<mint>({0, 1});
    }
    while ((int)buf.capacity() <= n) buf.reserve(buf.capacity() * 2);
    while ((int)buf.size() <= n) {
      long long k = buf.size(), q = (mod + k - 1) / k;
      buf.push_back(q * buf[k * q - mod]);
    }
    return buf[n];
  }
  static mint fact(int n) {
    static long long mod = mint::get_mod();
    static vector<mint> buf({1, 1});
    assert(n >= 0);
    if (mod != mint::get_mod()) {
      mod = mint::get_mod();
      buf = vector<mint>({1, 1});
    }
    while ((int)buf.capacity() <= n) buf.reserve(buf.capacity() * 2);
    while ((int)buf.size() <= n) {
      long long k = buf.size();
      buf.push_back(buf.back() * k);
    }
    return buf[n];
  }
  static mint fact_inv(int n) {
    static long long mod = mint::get_mod();
    static vector<mint> buf({1, 1});
    assert(n >= 0);
    if (mod != mint::get_mod()) {
      mod = mint::get_mod();
      buf = vector<mint>({1, 1});
    }
    if ((int)buf.size() <= n) inv(n);
    while ((int)buf.capacity() <= n) buf.reserve(buf.capacity() * 2);
    while ((int)buf.size() <= n) {
      long long k = buf.size();
      buf.push_back(buf.back() * inv(k));
    }
    return buf[n];
  }
  static mint binom(int n, int r) {
    if (r < 0 || r > n) return 0;
    return fact(n) * fact_inv(r) * fact_inv(n - r);
  }
  static mint binom_naive(int n, int r) {
    if (r < 0 || r > n) return 0;
    mint res = fact_inv(r);
    for (int i = 0; i < r; i++) res *= n - i;
    return res;
  }
  static mint multinom(const vector<int>& r) {
    int n = 0;
    for (auto& x : r) {
      if (x < 0) return 0;
      n += x;
    }
    mint res = fact(n);
    for (auto& x : r) res *= fact_inv(x);
    return res;
  }
  static mint P(int n, int r) {
    if (r < 0 || r > n) return 0;
    return fact(n) * fact_inv(n - r);
  }
  // partition n items to r groups (allow empty group)
  static mint H(int n, int r) {
    if (n < 0 || r < 0) return 0;
    return r == 0 ? 1 : binom(n + r - 1, r);
  }
};
/**
 * @brief 階乗, 二項係数
 */
#line 2 "fps/formal-power-series.hpp"

template <class mint>
struct FormalPowerSeries : vector<mint> {
  using vector<mint>::vector;
  using FPS = FormalPowerSeries;
  FPS &operator+=(const FPS &r) {
    if (r.size() > this->size()) this->resize(r.size());
    for (int i = 0; i < (int)r.size(); i++) (*this)[i] += r[i];
    return *this;
  }
  FPS &operator+=(const mint &r) {
    if (this->empty()) this->resize(1);
    (*this)[0] += r;
    return *this;
  }
  FPS &operator-=(const FPS &r) {
    if (r.size() > this->size()) this->resize(r.size());
    for (int i = 0; i < (int)r.size(); i++) (*this)[i] -= r[i];
    return *this;
  }
  FPS &operator-=(const mint &r) {
    if (this->empty()) this->resize(1);
    (*this)[0] -= r;
    return *this;
  }
  FPS &operator*=(const mint &v) {
    for (int k = 0; k < (int)this->size(); k++) (*this)[k] *= v;
    return *this;
  }
  FPS &operator/=(const FPS &r) {
    if (this->size() < r.size()) {
      this->clear();
      return *this;
    }
    int n = this->size() - r.size() + 1;
    if ((int)r.size() <= 64) {
      FPS f(*this), g(r);
      g.shrink();
      mint coeff = g.at(g.size() - 1).inv();
      for (auto &x : g) x *= coeff;
      int deg = (int)f.size() - (int)g.size() + 1;
      int gs = g.size();
      FPS quo(deg);
      for (int i = deg - 1; i >= 0; i--) {
        quo[i] = f[i + gs - 1];
        for (int j = 0; j < gs; j++) f[i + j] -= quo[i] * g[j];
      }
      *this = quo * coeff;
      this->resize(n, mint(0));
      return *this;
    }
    return *this = ((*this).rev().pre(n) * r.rev().inv(n)).pre(n).rev();
  }
  FPS &operator%=(const FPS &r) {
    *this -= *this / r * r;
    shrink();
    return *this;
  }
  FPS operator+(const FPS &r) const { return FPS(*this) += r; }
  FPS operator+(const mint &v) const { return FPS(*this) += v; }
  FPS operator-(const FPS &r) const { return FPS(*this) -= r; }
  FPS operator-(const mint &v) const { return FPS(*this) -= v; }
  FPS operator*(const FPS &r) const { return FPS(*this) *= r; }
  FPS operator*(const mint &v) const { return FPS(*this) *= v; }
  FPS operator/(const FPS &r) const { return FPS(*this) /= r; }
  FPS operator%(const FPS &r) const { return FPS(*this) %= r; }
  FPS operator-() const {
    FPS ret(this->size());
    for (int i = 0; i < (int)this->size(); i++) ret[i] = -(*this)[i];
    return ret;
  }
  void shrink() {
    while (this->size() && this->back() == mint(0)) this->pop_back();
  }
  FPS rev() const {
    FPS ret(*this);
    reverse(begin(ret), end(ret));
    return ret;
  }
  FPS dot(FPS r) const {
    FPS ret(min(this->size(), r.size()));
    for (int i = 0; i < (int)ret.size(); i++) ret[i] = (*this)[i] * r[i];
    return ret;
  }
  FPS pre(int sz) const {
    return FPS(begin(*this), begin(*this) + min((int)this->size(), sz));
  }
  FPS operator>>=(int sz) {
    assert(sz >= 0);
    if ((int)this->size() <= sz) return {};
    this->erase(this->begin(), this->begin() + sz);
    return *this;
  }
  FPS operator>>(int sz) const {
    if ((int)this->size() <= sz) return {};
    FPS ret(*this);
    ret.erase(ret.begin(), ret.begin() + sz);
    return ret;
  }
  FPS operator<<=(int sz) {
    assert(sz >= 0);
    this->insert(this->begin(), sz, mint(0));
    return *this;
  }
  FPS operator<<(int sz) const {
    FPS ret(*this);
    ret.insert(ret.begin(), sz, mint(0));
    return ret;
  }
  FPS diff() const {
    const int n = (int)this->size();
    FPS ret(max(0, n - 1));
    mint one(1), coeff(1);
    for (int i = 1; i < n; i++) {
      ret[i - 1] = (*this)[i] * coeff;
      coeff += one;
    }
    return ret;
  }
  FPS integral() const {
    const int n = (int)this->size();
    FPS ret(n + 1);
    ret[0] = mint(0);
    if (n > 0) ret[1] = mint(1);
    auto mod = mint::get_mod();
    for (int i = 2; i <= n; i++) ret[i] = (-ret[mod % i]) * (mod / i);
    for (int i = 0; i < n; i++) ret[i + 1] *= (*this)[i];
    return ret;
  }
  mint eval(mint x) const {
    mint r = 0, w = 1;
    for (auto &v : *this) r += w * v, w *= x;
    return r;
  }
  FPS log(int deg = -1) const {
    assert((*this)[0] == mint(1));
    if (deg == -1) deg = (int)this->size();
    return (this->diff() * this->inv(deg)).pre(deg - 1).integral();
  }
  FPS pow(int64_t k, int deg = -1) const {
    const int n = (int)this->size();
    if (deg == -1) deg = n;
    if (k == 0) {
      FPS ret(deg);
      if (deg) ret[0] = 1;
      return ret;
    }
    for (int i = 0; i < n; i++) {
      if ((*this)[i] != mint(0)) {
        mint rev = mint(1) / (*this)[i];
        FPS ret = (((*this * rev) >> i).log(deg) * k).exp(deg);
        ret *= (*this)[i].pow(k);
        ret = (ret << (i * k)).pre(deg);
        if ((int)ret.size() < deg) ret.resize(deg, mint(0));
        return ret;
      }
      if (__int128_t(i + 1) * k >= deg) return FPS(deg, mint(0));
    }
    return FPS(deg, mint(0));
  }

  static void *ntt_ptr;
  static void set_ntt();
  FPS &operator*=(const FPS &r);
  FPS middle_product(const FPS &r) const;
  void ntt();
  void intt();
  void ntt_doubling();
  static int ntt_root();
  FPS inv(int deg = -1) const;
  FPS exp(int deg = -1) const;
};
template <typename mint>
void *FormalPowerSeries<mint>::ntt_ptr = nullptr;
#line 4 "fps/prefix-sum-of-polynomial.hpp"

// g(n)=sum_{i=0}^{n}f(i)
template <class mint>
FormalPowerSeries<mint> PrefixSumOfPolynomial(FormalPowerSeries<mint> f) {
  if (f.empty()) return {};
  using fact = Factorial<mint>;
  mint c = f[0];
  int d = f.size();
  fact::reserve(d);
  for (int i = 0; i < d; i++) f[i] *= fact::fact(i);
  reverse(f.begin(), f.end());
  FormalPowerSeries<mint> g(d);
  for (int i = 0; i < d; i++) g[i] = fact::fact_inv(i + 1) * (i & 1 ? -1 : 1);
  f *= g.inv();
  f.resize(d);
  f.push_back(c);
  reverse(f.begin(), f.end());
  for (int i = 1; i <= d; i++) f[i] *= fact::fact_inv(i);
  return f;
}
/**
 * @brief 多項式の Prefix Sum
 * @docs docs/fps/prefix-sum-of-polynomial.md
 */

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