第二種スターリング数
(library/polynomial/fps/stirling_second_number.hpp)

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• Last update: 2026-01-21 11:49:22+09:00
• Include: #include "library/polynomial/fps/stirling_second_number.hpp"

第二種スターリング数

できること

• 入力Nに対し、 $k (0 \le k \le N)$についての第二種スターリング数 $\lbrace {}^n_k \rbrace$ を求める
• 戻り値のFPSの各次数がkに対応する
• 組み合わせ的に、区別できるN個のボールを区別できないK個の箱に、空箱が無いように分ける方法の数
  • ${}_n C_k$ ではないことに注意
• $\lbrace {}^n_k \rbrace$ は $x^n$ を下降階乗になおす際の係数として現れる
  • $x^n = \displaystyle\sum_{k=0}^{n} \lbrace {}^n_k \rbrace x(x-1) \cdots (x-k+1)$
• よって、ファウルハーバーの公式の別バージョンとしてベルヌーイ数でなく、第二種スターリング数を用いたべき乗和の公式が存在する

計算量

$O(NlogN)$

使い方

using mint = modint998244353;
cout << stirling_second_number<FPS, mint>(N);

Depends on

• モジュロ演算 (library/number/mod/montgomery_mod_int.hpp)
• 畳み込み 任意MOD (library/polynomial/fft/convolution_arbitrary_mod.hpp)
• 高速フーリエ変換 (library/polynomial/fft/fast_fourier_transform.hpp)
• 形式的冪級数 (library/polynomial/fps/formal_power_series.hpp)

Verified with

• 第二種スターリング数のテスト (tests/polynomial.fps.stirling_second_number.test.cpp)

Code

#pragma once
#include "library/polynomial/fps/formal_power_series.hpp"
template <template <typename> class FPS, typename Mint>
FPS<Mint> stirling_second_number(int N) {
    FPS<Mint> A(N + 1), B(N + 1);
    Mint tmp = 1;
    for (int i = 0; i <= N; i++) {
        Mint rev = Mint(1) / tmp;
        A[i] = Mint(i).pow(N) * rev;
        B[i] = Mint(1) * rev;
        if (i & 1) B[i] *= -1;
        tmp *= i + 1;
    }
    return (A * B).pre(N + 1);
}

#line 2 "library/polynomial/fps/formal_power_series.hpp"
#include <functional>
#line 2 "library/number/mod/montgomery_mod_int.hpp"
template <uint32_t mod_, bool fast = false> struct MontgomeryModInt {
  private:
    using mint = MontgomeryModInt;
    using i32 = int32_t;
    using i64 = int64_t;
    using u32 = uint32_t;
    using u64 = uint64_t;
    static constexpr u32 get_r() {
        u32 ret = mod_;
        for (i32 i = 0; i < 4; i++) ret *= 2 - mod_ * ret;
        return ret;
    }
    static constexpr u32 r = get_r();
    static constexpr u32 n2 = -u64(mod_) % mod_;
    static_assert(r * mod_ == 1, "invalid, r * mod != 1");
    static_assert(mod_ < (1 << 30), "invalid, mod >= 2 ^ 30");
    static_assert((mod_ & 1) == 1, "invalid, mod % 2 == 0");
    u32 x;

  public:
    MontgomeryModInt() : x{} {}
    MontgomeryModInt(const i64 &a)
        : x(reduce(u64(fast ? a : (a % mod() + mod())) * n2)) {}
    static constexpr u32 reduce(const u64 &b) {
        return u32(b >> 32) + mod() - u32((u64(u32(b) * r) * mod()) >> 32);
    }
    mint &operator+=(const mint &p) {
        if (i32(x += p.x - 2 * mod()) < 0) x += 2 * mod();
        return *this;
    }
    mint &operator-=(const mint &p) {
        if (i32(x -= p.x) < 0) x += 2 * mod();
        return *this;
    }
    mint &operator*=(const mint &p) {
        x = reduce(u64(x) * p.x);
        return *this;
    }
    mint &operator/=(const mint &p) {
        *this *= p.inv();
        return *this;
    }
    mint operator-() const { return mint() - *this; }
    mint operator+(const mint &p) const { return mint(*this) += p; }
    mint operator-(const mint &p) const { return mint(*this) -= p; }
    mint operator*(const mint &p) const { return mint(*this) *= p; }
    mint operator/(const mint &p) const { return mint(*this) /= p; }
    bool operator==(const mint &p) const {
        return (x >= mod() ? x - mod() : x) ==
               (p.x >= mod() ? p.x - mod() : p.x);
    }
    bool operator!=(const mint &p) const {
        return (x >= mod() ? x - mod() : x) !=
               (p.x >= mod() ? p.x - mod() : p.x);
    }
    u32 val() const {
        u32 ret = reduce(x);
        return ret >= mod() ? ret - mod() : ret;
    }
    mint pow(u64 n) const {
        mint ret(1), mul(*this);
        while (n > 0) {
            if (n & 1) ret *= mul;
            mul *= mul;
            n >>= 1;
        }
        return ret;
    }
    mint inv() const { return pow(mod() - 2); }
    friend ostream &operator<<(ostream &os, const mint &p) {
        return os << p.val();
    }
    friend istream &operator>>(istream &is, mint &a) {
        i64 t;
        is >> t;
        a = mint(t);
        return is;
    }
    static constexpr u32 mod() { return mod_; }
};
template <uint32_t mod> using modint = MontgomeryModInt<mod>;
using modint998244353 = modint<998244353>;
using modint1000000007 = modint<1000000007>;
#line 2 "library/polynomial/fft/fast_fourier_transform.hpp"
namespace FFT {
using real = double;
struct C {
    real x, y;
    C() : x(0), y(0) {};
    C(real x, real y) : x(x), y(y) {};
    inline C operator+(const C &c) const { return C(x + c.x, y + c.y); }
    inline C operator-(const C &c) const { return C(x - c.x, y - c.y); }
    inline C operator*(const C &c) const {
        return C(x * c.x - y * c.y, x * c.y + y * c.x);
    }
    inline C conj() const { return C(x, -y); }
};
const real PI = acosl(-1);
int base = 1;
vector<C> rts = {{0, 0}, {1, 0}};
vector<int> rev = {0, 1};
void ensure_base(int nbase) {
    if (nbase <= base) return;
    rev.resize(1 << nbase);
    rts.resize(1 << nbase);
    for (int i = 0; i < (1 << nbase); i++) {
        rev[i] = (rev[i >> 1] >> 1) + ((i & 1) << (nbase - 1));
    }
    while (base < nbase) {
        real angle = PI * 2.0 / (1 << (base + 1));
        for (int i = 1 << (base - 1); i < (1 << base); i++) {
            rts[i << 1] = rts[i];
            real angle_i = angle * (2 * i + 1 - (1 << base));
            rts[(i << 1) + 1] = C(cos(angle_i), sin(angle_i));
        }
        ++base;
    }
}
void fft(vector<C> &a, int n) {
    assert((n & (n - 1)) == 0);
    int zeros = __builtin_ctz(n);
    ensure_base(zeros);
    int shift = base - zeros;
    for (int i = 0; i < n; i++) {
        if (i < (rev[i] >> shift)) swap(a[i], a[rev[i] >> shift]);
    }
    for (int k = 1; k < n; k <<= 1) {
        for (int i = 0; i < n; i += 2 * k) {
            for (int j = 0; j < k; j++) {
                C z = a[i + j + k] * rts[j + k];
                a[i + j + k] = a[i + j] - z;
                a[i + j] = a[i + j] + z;
            }
        }
    }
}
vector<int64_t> multiply(const vector<int> &a, const vector<int> &b) {
    int need = (int)a.size() + (int)b.size() - 1;
    int nbase = 1;
    while ((1 << nbase) < need) ++nbase;
    ensure_base(nbase);
    int sz = 1 << nbase;
    vector<C> fa(sz);
    for (int i = 0; i < sz; ++i) {
        int x = (i < (int)a.size() ? a[i] : 0);
        int y = (i < (int)b.size() ? b[i] : 0);
        fa[i] = C(x, y);
    }
    fft(fa, sz);
    C r(0, -0.25 / (sz >> 1)), s(0, 1), t(0.5, 0);
    for (int i = 0; i <= (sz >> 1); i++) {
        int j = (sz - i) & (sz - 1);
        C z = (fa[j] * fa[j] - (fa[i] * fa[i]).conj()) * r;
        fa[j] = (fa[i] * fa[i] - (fa[j] * fa[j]).conj()) * r;
        fa[i] = z;
    }
    for (int i = 0; i < (sz >> 1); i++) {
        C A0 = (fa[i] + fa[i + (sz >> 1)]) * t;
        C A1 = (fa[i] - fa[i + (sz >> 1)]) * t * rts[(sz >> 1) + i];
        fa[i] = A0 + A1 * s;
    }
    fft(fa, sz >> 1);
    vector<int64_t> ret(need);
    for (int i = 0; i < need; i++) {
        ret[i] = llround(i & 1 ? fa[i >> 1].y : fa[i >> 1].x);
    }
    return ret;
}
} // namespace FFT
#line 4 "library/polynomial/fft/convolution_arbitrary_mod.hpp"
template <typename T> struct ConvolutionArbitraryMod {
    using real = FFT::real;
    using C = FFT::C;
    ConvolutionArbitraryMod() = default;
    static vector<T> multiply(const vector<T> &a, const vector<T> &b,
                              int need = -1) {
        if (need == -1) need = a.size() + b.size() - 1;
        int nbase = 0;
        while ((1 << nbase) < need) nbase++;
        FFT::ensure_base(nbase);
        int sz = 1 << nbase;
        vector<C> fa(sz);
        for (int i = 0; i < (int)a.size(); i++) {
            fa[i] = C(a[i].val() & ((1 << 15) - 1), a[i].val() >> 15);
        }
        fft(fa, sz);
        vector<C> fb(sz);
        if (a == b) {
            fb = fa;
        } else {
            for (int i = 0; i < (int)b.size(); i++) {
                fb[i] = C(b[i].val() & ((1 << 15) - 1), b[i].val() >> 15);
            }
            fft(fb, sz);
        }
        real ratio = 0.25 / sz;
        C r2(0, -1), r3(ratio, 0), r4(0, -ratio), r5(0, 1);
        for (int i = 0; i <= (sz >> 1); i++) {
            int j = (sz - i) & (sz - 1);
            C a1 = (fa[i] + fa[j].conj());
            C a2 = (fa[i] - fa[j].conj()) * r2;
            C b1 = (fb[i] + fb[j].conj()) * r3;
            C b2 = (fb[i] - fb[j].conj()) * r4;
            if (i != j) {
                C c1 = (fa[j] + fa[i].conj());
                C c2 = (fa[j] - fa[i].conj()) * r2;
                C d1 = (fb[j] + fb[i].conj()) * r3;
                C d2 = (fb[j] - fb[i].conj()) * r4;
                fa[i] = c1 * d1 + c2 * d2 * r5;
                fb[i] = c1 * d2 + c2 * d1;
            }
            fa[j] = a1 * b1 + a2 * b2 * r5;
            fb[j] = a1 * b2 + a2 * b1;
        }
        fft(fa, sz);
        fft(fb, sz);
        vector<T> ret(need);
        for (int i = 0; i < need; i++) {
            int64_t aa = llround(fa[i].x);
            int64_t bb = llround(fb[i].x);
            int64_t cc = llround(fa[i].y);
            aa = T(aa).val(), bb = T(bb).val(), cc = T(cc).val();
            ret[i] = aa + (bb << 15) + (cc << 30);
        }
        return ret;
    }
};
#line 4 "library/polynomial/fps/formal_power_series.hpp"
template <typename T> struct FormalPowerSeries : vector<T> {
    using vector<T>::vector;
    using P = FormalPowerSeries;
    using Conv = ConvolutionArbitraryMod<T>;
    P pre(int deg) const {
        return P(begin(*this), begin(*this) + min((int)this->size(), deg));
    }
    P rev(int deg = -1) const {
        P ret(*this);
        if (deg != -1) ret.resize(deg, T(0));
        reverse(begin(ret), end(ret));
        return ret;
    }
    void shrink() {
        while (this->size() && this->back() == T(0)) this->pop_back();
    }
    P operator+(const P &r) const { return P(*this) += r; }
    P operator+(const T &v) const { return P(*this) += v; }
    P operator-(const P &r) const { return P(*this) -= r; }
    P operator-(const T &v) const { return P(*this) -= v; }
    P operator*(const P &r) const { return P(*this) *= r; }
    P operator*(const T &v) const { return P(*this) *= v; }
    P operator/(const P &r) const { return P(*this) /= r; }
    P operator%(const P &r) const { return P(*this) %= r; }
    P &operator+=(const P &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;
    }
    P &operator-=(const P &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;
    }
    // https://judge.yosupo.jp/problem/convolution_mod
    P &operator*=(const P &r) {
        if (this->empty() || r.empty()) {
            this->clear();
            return *this;
        }
        auto ret = Conv::multiply(*this, r);
        return *this = {begin(ret), end(ret)};
    }
    P &operator/=(const P &r) {
        if (this->size() < r.size()) {
            this->clear();
            return *this;
        }
        int n = this->size() - r.size() + 1;
        return *this = (rev().pre(n) * r.rev().inv(n)).pre(n).rev(n);
    }
    P &operator%=(const P &r) { return *this -= *this / r * r; }
    // https://judge.yosupo.jp/problem/division_of_polynomials
    pair<P, P> div_mod(const P &r) {
        P q = *this / r;
        return make_pair(q, *this - q * r);
    }
    P operator-() const {
        P ret(this->size());
        for (int i = 0; i < (int)this->size(); i++) ret[i] = -(*this)[i];
        return ret;
    }
    P &operator+=(const T &r) {
        if (this->empty()) this->resize(1);
        (*this)[0] += r;
        return *this;
    }
    P &operator-=(const T &r) {
        if (this->empty()) this->resize(1);
        (*this)[0] -= r;
        return *this;
    }
    P &operator*=(const T &v) {
        for (int i = 0; i < (int)this->size(); i++) (*this)[i] *= v;
        return *this;
    }
    P dot(P r) const {
        P ret(min(this->size(), r.size()));
        for (int i = 0; i < (int)ret.size(); i++) ret[i] = (*this)[i] * r[i];
        return ret;
    }
    P operator>>(int sz) const {
        if (this->size() <= sz) return {};
        P ret(*this);
        ret.erase(ret.begin(), ret.begin() + sz);
        return ret;
    }
    P operator<<(int sz) const {
        P ret(*this);
        ret.insert(ret.begin(), sz, T(0));
        return ret;
    }
    T operator()(T x) const {
        T r = 0, w = 1;
        for (auto &v : *this) {
            r += w * v;
            w *= x;
        }
        return r;
    }
    P diff() const {
        const int n = (int)this->size();
        P ret(max(0ll, n - 1));
        for (int i = 1; i < n; i++) ret[i - 1] = (*this)[i] * T(i);
        return ret;
    }
    P integral() const {
        const int n = (int)this->size();
        P ret(n + 1);
        ret[0] = T(0);
        for (int i = 0; i < n; i++) ret[i + 1] = (*this)[i] / T(i + 1);
        return ret;
    }
    // https://judge.yosupo.jp/problem/inv_of_formal_power_series
    // F(0) must not be 0
    P inv(int deg = -1) const {
        assert(((*this)[0]) != T(0));
        const int n = (int)this->size();
        if (deg == -1) deg = n;
        P ret({T(1) / (*this)[0]});
        for (int i = 1; i < deg; i <<= 1) {
            ret = (ret + ret - ret * ret * pre(i << 1)).pre(i << 1);
        }
        return ret.pre(deg);
    }
    // https://judge.yosupo.jp/problem/log_of_formal_power_series
    // F(0) must be 1
    P log(int deg = -1) const {
        assert((*this)[0] == T(1));
        const int n = (int)this->size();
        if (deg == -1) deg = n;
        return (this->diff() * this->inv(deg)).pre(deg - 1).integral();
    }
    // https://judge.yosupo.jp/problem/sqrt_of_formal_power_series
    P sqrt(
        int deg = -1,
        const function<T(T)> &get_sqrt = [](T) { return T(1); }) const {
        const int n = (int)this->size();
        if (deg == -1) deg = n;
        if ((*this)[0] == T(0)) {
            for (int i = 1; i < n; i++) {
                if ((*this)[i] != T(0)) {
                    if (i & 1) return {};
                    if (deg - i / 2 <= 0) break;
                    auto ret = (*this >> i).sqrt(deg - i / 2, get_sqrt);
                    if (ret.empty()) return {};
                    ret = ret << (i / 2);
                    if (ret.size() < deg) ret.resize(deg, T(0));
                    return ret;
                }
            }
            return P(deg, 0);
        }
        auto sqr = T(get_sqrt((*this)[0]));
        if (sqr * sqr != (*this)[0]) return {};
        P ret{sqr};
        T inv2 = T(1) / T(2);
        for (int i = 1; i < deg; i <<= 1) {
            ret = (ret + pre(i << 1) * ret.inv(i << 1)) * inv2;
        }
        return ret.pre(deg);
    }
    P sqrt(const function<T(T)> &get_sqrt, int deg = -1) const {
        return sqrt(deg, get_sqrt);
    }
    // https://judge.yosupo.jp/problem/exp_of_formal_power_series
    // F(0) must be 0
    P exp(int deg = -1) const {
        if (deg == -1) deg = this->size();
        assert((*this)[0] == T(0));
        const int n = (int)this->size();
        if (deg == -1) deg = n;
        P ret({T(1)});
        for (int i = 1; i < deg; i <<= 1) {
            ret = (ret * (pre(i << 1) + T(1) - ret.log(i << 1))).pre(i << 1);
        }
        return ret.pre(deg);
    }
    // https://judge.yosupo.jp/problem/pow_of_formal_power_series
    P pow(int64_t k, int deg = -1) const {
        const int n = (int)this->size();
        if (deg == -1) deg = n;
        if (k == 0) {
            P ret(deg, T(0));
            ret[0] = T(1);
            return ret;
        }
        for (int i = 0; i < n; i++) {
            if (i * k > deg) return P(deg, T(0));
            if ((*this)[i] != T(0)) {
                T rev = T(1) / (*this)[i];
                P ret = (((*this * rev) >> i).log() * k).exp() *
                        ((*this)[i].pow(k));
                ret = (ret << (i * k)).pre(deg);
                if (ret.size() < deg) ret.resize(deg, T(0));
                return ret;
            }
        }
        return *this;
    }
    // https://yukicoder.me/problems/no/215
    P mod_pow(int64_t k, P g) const {
        P modinv = g.rev().inv();
        auto get_div = [&](P base) {
            if (base.size() < g.size()) {
                base.clear();
                return base;
            }
            int n = base.size() - g.size() + 1;
            return (base.rev().pre(n) * modinv.pre(n)).pre(n).rev(n);
        };
        P x(*this), ret{1};
        while (k > 0) {
            if (k & 1) {
                ret *= x;
                ret -= get_div(ret) * g;
                ret.shrink();
            }
            x *= x;
            x -= get_div(x) * g;
            x.shrink();
            k >>= 1;
        }
        return ret;
    }
    // https://judge.yosupo.jp/problem/polynomial_taylor_shift
    P taylor_shift(T c) const {
        int n = (int)this->size();
        vector<T> fact(n), rfact(n);
        fact[0] = rfact[0] = T(1);
        for (int i = 1; i < n; i++) fact[i] = fact[i - 1] * T(i);
        rfact[n - 1] = T(1) / fact[n - 1];
        for (int i = n - 1; i > 1; i--) rfact[i - 1] = rfact[i] * T(i);
        P p(*this);
        for (int i = 0; i < n; i++) p[i] *= fact[i];
        p = p.rev();
        P bs(n, T(1));
        for (int i = 1; i < n; i++)
            bs[i] = bs[i - 1] * c * rfact[i] * fact[i - 1];
        p = (p * bs).pre(n);
        p = p.rev();
        for (int i = 0; i < n; i++) p[i] *= rfact[i];
        return p;
    }
};
template <typename Mint> using FPS = FormalPowerSeries<Mint>;
#line 3 "library/polynomial/fps/stirling_second_number.hpp"
template <template <typename> class FPS, typename Mint>
FPS<Mint> stirling_second_number(int N) {
    FPS<Mint> A(N + 1), B(N + 1);
    Mint tmp = 1;
    for (int i = 0; i <= N; i++) {
        Mint rev = Mint(1) / tmp;
        A[i] = Mint(i).pow(N) * rev;
        B[i] = Mint(1) * rev;
        if (i & 1) B[i] *= -1;
        tmp *= i + 1;
    }
    return (A * B).pre(N + 1);
}

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