LibreOJ 3397 「2020-2021 集训队作业」春天，在积雪下结一成形，抽枝发芽

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首先特判一下 \(n\le 2\) 的情况。

设 \(H(x)\) 表示排列的 OGF（此处令 \(0!=0\)），\(G(x)\) 表示根结点为合点且儿子序列为升序的排列个数。
那么有 \[ G(x) = \sum\limits_{i=2}^{\infty} (H(x)-G(x))^i \]

因此有 \[ G(x) = \frac{H^2(x)}{H(x)+1} \]

然后设 \(F(x)\) 表示答案，那么易知根结点为析点的排列个数即为 \[ \sum\limits_{i=4}^{\infty} f_i H^i(x) = F(H(x)) \]

那么有 \[ F(H(x)) + 2G(x) + x = H(x) \]

设 \(P(x) = H(x) - 2G(x) - x\)，那么有 \[ F(H(x)) = P(x) \]

代入 \(H^{-1}(x)\) 可得 \[ F(x) = P(H^{-1}(x)) = x - \frac{2x^2}{1+x} - H^{-1}(x) \]

于是问题瓶颈变为求 \(H^{-1}\)。

令 \(A = H^{-1}\)。
观察 \(H\) 易得 \[ H(x) = x^2H'(x) + xH(x) + x \]

代入 \(A\) 得 \[ \begin{aligned} \frac{A^2(x)}{A'(x)} + (x + 1)A(x) &= x \\ A^2(x) + (x+1)A(x)A'(x) - xA'(x) &= 0 \end{aligned} \]

从而可以得到递推式 \[ a_i = \begin{cases} 0, & i = 0 \\ 1, & i = 1 \\ -\sum_{j=1}^{i-1} (j+1) a_j a_{i-j} - \sum_{j=2}^{i-1} j a_j a_{i-j+1}, & i \ge 2 \end{cases} \]

复杂度 \(O(n \log^2 n)\)。
懒得卡常了。

代码： 

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#include <cstdio>
#include <vector>
#include <cstring>
#include <algorithm>
#define add(a,b) (a + b >= mod ? a + b - mod : a + b)
#define dec(a,b) (a < b ? a - b + mod : a - b)
using namespace std;
const int N = 1e5;
const int mod = 998244353;
inline int fpow(int a,int b)
{
    int ret = 1;
    for(;b;b >>= 1)
        (b & 1) && (ret = (long long)ret * a % mod),a = (long long)a * a % mod;
    return ret;
}
int type,n;
namespace Poly
{
    const int N = 1 << 18;
    const int G = 3;
    int lg2[N + 5];
    int rev[N + 5],fac[N + 5],ifac[N + 5],inv[N + 5];
    int rt[N + 5],irt[N + 5];
    inline void init()
    {
        for(register int i = 2;i <= N;++i)
            lg2[i] = lg2[i >> 1] + 1;
        int w = fpow(G,(mod - 1) / N);
        rt[N >> 1] = 1;
        for(register int i = (N >> 1) + 1;i <= N;++i)
            rt[i] = (long long)rt[i - 1] * w % mod;
        for(register int i = (N >> 1) - 1;i;--i)
            rt[i] = rt[i << 1];
        fac[0] = 1;
        for(register int i = 1;i <= N;++i)
            fac[i] = (long long)fac[i - 1] * i % mod;
        ifac[N] = fpow(fac[N],mod - 2);
        for(register int i = N;i;--i)
            ifac[i - 1] = (long long)ifac[i] * i % mod;
        for(register int i = 1;i <= N;++i)
            inv[i] = (long long)ifac[i] * fac[i - 1] % mod;
    }
    struct poly
    {
        vector<int> a;
        inline poly(int x = 0)
        {
            x && (a.push_back(x),1);
        }
        inline poly(const vector<int> &o)
        {
            a = o,shrink();
        }
        inline poly(const poly &o)
        {
            a = o.a,shrink();
        }
        inline void shrink()
        {
            for(;!a.empty() && !a.back();a.pop_back());
        }
        inline int size() const
        {
            return a.size();
        }
        inline void resize(int x)
        {
            a.resize(x);
        }
        inline int operator[](int x) const
        {
            if(x < 0 || x >= size())
                return 0;
            return a[x];
        }
        inline void clear()
        {
            vector<int>().swap(a);
        }
        inline poly rever() const
        {
            return poly(vector<int>(a.rbegin(),a.rend()));
        }
        inline void ntt(int type = 1)
        {
            int n = size();
            type == -1 && (reverse(a.begin() + 1,a.end()),1);
            int lg = lg2[n] - 1;
            for(register int i = 0;i < n;++i)
                rev[i] = (rev[i >> 1] >> 1) | ((i & 1) << lg),
                i < rev[i] && (swap(a[i],a[rev[i]]),1);
            for(register int w = 2,m = 1;w <= n;w <<= 1,m <<= 1)
                for(register int i = 0;i < n;i += w)
                    for(register int j = 0;j < m;++j)
                    {
                        int t = (long long)rt[m | j] * a[i | j | m] % mod;
                        a[i | j | m] = dec(a[i | j],t),a[i | j] = add(a[i | j],t);
                    }
            if(type == -1)
                for(register int i = 0;i < n;++i)
                    a[i] = (long long)a[i] * inv[n] % mod;
        }
        friend inline poly operator+(const poly &a,const poly &b)
        {
            vector<int> ret(max(a.size(),b.size()));
            for(register int i = 0;i < ret.size();++i)
                ret[i] = add(a[i],b[i]);
            return poly(ret);
        }
        friend inline poly operator-(const poly &a,const poly &b)
        {
            vector<int> ret(max(a.size(),b.size()));
            for(register int i = 0;i < ret.size();++i)
                ret[i] = dec(a[i],b[i]);
            return poly(ret);
        }
        friend inline poly operator*(poly a,poly b)
        {
            if(a.a.empty() || b.a.empty())
                return poly();
            int lim = 1,tot = a.size() + b.size() - 1;
            for(;lim < tot;lim <<= 1);
            a.resize(lim),b.resize(lim);
            a.ntt(),b.ntt();
            for(register int i = 0;i < lim;++i)
                a.a[i] = (long long)a[i] * b[i] % mod;
            a.ntt(-1),a.shrink();
            return a;
        }
        poly &operator+=(const poly &o)
        {
            resize(max(size(),o.size()));
            for(register int i = 0;i < o.size();++i)
                a[i] = add(a[i],o[i]);
            return *this;
        }
        poly &operator-=(const poly &o)
        {
            resize(max(size(),o.size()));
            for(register int i = 0;i < o.size();++i)
                a[i] = dec(a[i],o[i]);
            return *this;
        }
        poly &operator*=(poly o)
        {
            return (*this) = (*this) * o;
        }
        poly deriv() const
        {
            if(a.empty())
                return poly();
            vector<int> ret(size() - 1);
            for(register int i = 0;i < size() - 1;++i)
                ret[i] = (long long)(i + 1) * a[i + 1] % mod;
            return poly(ret);
        }
        poly integ() const
        {
            if(a.empty())
                return poly();
            vector<int> ret(size() + 1);
            for(register int i = 0;i < size();++i)
                ret[i + 1] = (long long)a[i] * inv[i + 1] % mod;
            return poly(ret);
        }
        inline poly modxn(int n) const
        {
            if(a.empty())
                return poly();
            n = min(n,size());
            return poly(vector<int>(a.begin(),a.begin() + n));
        }
        inline poly inver(int m) const
        {
            poly ret(fpow(a[0],mod - 2));
            for(register int k = 1;k < m;)
                k <<= 1,ret = (ret * (2 - modxn(k) * ret)).modxn(k);
            return ret.modxn(m);
        }
        inline pair<poly,poly> div(poly o) const
        {
            if(size() < o.size())
                return make_pair(poly(),*this);
            poly f,g;
            f = (rever().modxn(size() - o.size() + 1) * o.rever().inver(size() - o.size() + 1)).modxn(size() - o.size() + 1).rever();
            g = (modxn(o.size() - 1) - o.modxn(o.size() - 1) * f.modxn(o.size() - 1)).modxn(o.size() - 1);
            return make_pair(f,g);
        }
        inline poly log(int m) const
        {
            return (deriv() * inver(m)).integ().modxn(m);
        }
        inline poly exp(int m) const
        {
            poly ret(1);
            for(register int k = 1;k < m;)
                k <<= 1,ret = (ret * (1 - ret.log(k) + modxn(k))).modxn(k);
            return ret.modxn(m);
        }
        inline poly pow(int m,int k1,int k2 = -1) const
        {
            if(a.empty())
                return poly();
            if(k2 == -1)
                k2 = k1;
            int t = 0;
            for(;t < size() && !a[t];++t);
            if((long long)t * k1 >= m)
                return poly();
            poly ret;
            ret.resize(m);
            int u = fpow(a[t],mod - 2),v = fpow(a[t],k2);
            for(register int i = 0;i < m - t * k1;++i)
                ret.a[i] = (long long)operator[](i + t) * u % mod;
            ret = ret.log(m - t * k1);
            for(register int i = 0;i < ret.size();++i)
                ret.a[i] = (long long)ret[i] * k1 % mod;
            ret = ret.exp(m - t * k1),t *= k1,ret.resize(m);
            for(register int i = m - 1;i >= t;--i)
                ret.a[i] = (long long)ret[i - t] * v % mod;
            for(register int i = 0;i < t;++i)
                ret.a[i] = 0;
            return ret;
        }
    };
}
using Poly::init;
using Poly::poly;
int f[N + 5];
poly a,b;
void solve(int l,int r)
{
    if(l == r)
        return ;
    int mid = l + r >> 1;
    solve(l,mid);
    if(l > 1)
    {
        a.resize(mid - l + 1),b.resize(r - l + 1);
        for(register int i = 0;i <= mid - l;++i)
            a.a[i] = (long long)(i + l + 1) * f[i + l] % mod;
        for(register int i = 0;i <= r - l;++i)
            b.a[i] = f[i];
        a *= b;
        for(register int i = mid + 1;i <= r;++i)
            f[i] = dec(f[i],a[i - l]);
        a.resize(mid - l + 1),b.resize(r - l + 1);
        for(register int i = 0;i <= mid - l;++i)
            a.a[i] = f[i + l];
        for(register int i = 0;i <= r - l;++i)
            b.a[i] = (long long)(i + 1) * f[i] % mod;
        a *= b;
        for(register int i = mid + 1;i <= r;++i)
            f[i] = dec(f[i],a[i - l]);
        a.resize(mid - l + 1),b.resize(r - l + 2);
        for(register int i = 0;i <= mid - l;++i)
            a.a[i] = (long long)(i + l) * f[i + l] % mod;
        for(register int i = 0;i <= r - l + 1;++i)
            b.a[i] = f[i];
        a *= b;
        for(register int i = mid + 1;i <= r;++i)
            f[i] = dec(f[i],a[i - l + 1]);
        a.resize(mid - l + 1),b.resize(r - l + 2);
        for(register int i = 0;i <= mid - l;++i)
            a.a[i] = f[i + l];
        for(register int i = 0;i <= r - l + 1;++i)
            b.a[i] = (long long)i * f[i] % mod;
        a *= b;
        for(register int i = mid + 1;i <= r;++i)
            f[i] = dec(f[i],a[i - l + 1]);
    }
    else
    {
        a.resize(mid + 1),b.resize(mid + 1);
        for(register int i = 0;i <= mid;++i)
            a.a[i] = (long long)(i + 1) * f[i] % mod,
            b.a[i] = f[i];
        a *= b;
        for(register int i = mid + 1;i <= r;++i)
            f[i] = dec(f[i],a[i]);
        a.resize(mid + 1),b.resize(mid + 1);
        for(register int i = 0;i <= mid;++i)
            a.a[i] = (long long)i * f[i] % mod,
            b.a[i] = f[i];
        a *= b;
        for(register int i = mid + 1;i <= r;++i)
            f[i] = dec(f[i],a[i + 1]);
    }
    solve(mid + 1,r);
}
int main()
{
    Poly::init();
    scanf("%d%d",&type,&n),f[1] = 1;
    solve(1,n);
    for(register int i = 1;i <= n;++i)
        f[i] = (mod - f[i]) % mod,
        f[i] = i & 1 ? add(f[i],2) : dec(f[i],2);
    f[2] = 2;
    for(register int i = 1;i <= n;++i)
        (type || i == n) && printf("%d\n",f[i]);
}