[BZOJ3625][Codeforces Round #250]小朋友和二叉树 多项式开根+求逆

https://www.lydsy.com/JudgeOnline/problem.php?id=3625

愉快地列式子。设\(F[i]\)表示权值为\(i\) 的子树的方案数，\(A[i]\)为\(i\)在不在集合中。
\[ F[n]=\sum_{i=0}^n \sum_{j=0}^{n-i}F[i]\cdot F[j]\cdot A[n-i-j] \]
初始状态\(F[0]=1\)。

我们把\(F,A\)看成多项式。
\[ F(x)-1=F^2(x)\cdot A(x)\\ A(x)\cdot F^2(x)-F(x)+1=0\\ F(x)=\frac{1\pm\sqrt{1-4A(x)}}{2A(x)} \]
因为\(A[0]=0\)而\(F[0]=1\)，如果取\(+\)号，末位会不符，舍出。因此只能取\(-\)。

这样
\[ \begin{align*} F(x)&=\frac{1-\sqrt{1-4A(x)}}{2A(x)}\\ &=\frac{(1-\sqrt{1-4A(x)})(1+\sqrt{1-4A(x)})}{2A(x)(1+\sqrt{1-4A(x)})}\\ &=\frac{4A(x)}{2A(x)(1+\sqrt{1-4A(x)})}\\ &=\frac{2}{1+\sqrt{1-4A(x)}} \end{align*} \]
这样就变成多项式开根+求逆的板子了。

---

(刚开始转码风，可能有些地方不太自然，也有可能有些地方仍然保留着就码风没有注意)

#include<bits/stdc++.h>

#define fec(i, x, y) (int i = head[x], y = g[i].to; i; i = g[i].ne, y = g[i].to)
#define dbg(...) fprintf(stderr, __VA_ARGS__)
#define File(x) freopen(#x".in", "r", stdin), freopen(#x".out", "w", stdout)
#define fi first
#define se second
#define pb push_back
#define isbreak dbg("*")

template<typename A, typename B> inline char SMAX(A &a, const B &b) {return a < b ? a = b , 1 : 0;}
template<typename A, typename B> inline char SMIN(A &a, const B &b) {return b < a ? a = b , 1 : 0;}

typedef long long ll; typedef unsigned long long ull; typedef std::pair<int, int> pii;

template<typename I>
inline void read(I &x) {
    int f = 0, c;
    while (!isdigit(c = getchar())) c == '-' ? f = 1 : 0;
    x = c & 15;
    while (isdigit(c = getchar())) x = (x << 1) + (x << 3) + (c & 15);
    f ? x = -x : 0;
}

const int N = 4e5 + 7;
const int P = 998244353;
const int G = 3;
const int Gi = 332748118;
const int Inv2 = 499122177;

int n, m, x, a[N];

inline void SADD(int &x, int y) {x += y;x >= P ? x -= P : 0;}
inline int SMOD(int x) {return x >= P ? x - P : x;}
inline int fpow(int x,int y) {
    int ans = 1;
    for (; y; y >>= 1, x = (ll)x * x % P) if (y & 1) ans = (ll)ans * x % P;
    return ans;
}

namespace DFT {
    int A[N], B[N], C[N];

    inline void NTT(int *a, int n, int f) {
        for (int i = 0, j = 0; i < n; ++i) {
            if (i > j) std::swap(a[i], a[j]);
            for (int l = n >> 1; (j ^= l) < l; l >>= 1);
        }
        for (int i = 1; i < n; i <<= 1) {
            int w = fpow(f > 0 ? G : Gi, (P - 1) / (i << 1));
            for (int j = 0; j < n; j += i << 1)
                for (int k = 0, e = 1; k < i; ++k, e = (ll)e * w % P){
                    int x = a[j + k], y = (ll)e * a[i + j + k] % P;
                    a[j + k] = SMOD(x + y); a[i + j + k] = SMOD(x + P - y);
                }
        }
        if (f < 0) for (int i = 0, p = fpow(n, P - 2); i < n; ++i) a[i] = (ll)a[i] * p % P;
    }
    namespace Inv {
        int A[N], B[N];
        inline void GetInv(int *a, int n, int *b) {
            memset(B, 0, sizeof(B)); B[0] = fpow(a[0], P - 2);
            for (int deg = 2; deg < (n << 1); deg <<= 1) {
                int L = deg << 1;
                for (int i = 0; i < deg; ++i) A[i] = a[i];
                for (int i = deg; i < L; ++i) A[i] = 0;
                NTT(A, L, 1); NTT(B, L, 1);
                for (int i = 0; i < L; ++i) B[i] = (ll)B[i] * (2 + P - (ll)B[i] * A[i] % P) % P;
                NTT(B, L, -1);
                for (int i = deg; i < L; ++i) B[i] = 0;
            }
            for (int i = 0; i < n; ++i) b[i] = B[i];
        }
    } using Inv::GetInv;

    inline void GetSqrt(int *a, int n, int *b) {
        B[0] = 1;
        for (int deg = 2; deg < (n << 1); deg <<= 1) {
            int L = deg << 1;
            for (int i = 0; i < deg; ++i) A[i] = a[i];
            for (int i = deg; i < L; ++i) A[i] = 0;
            GetInv(B, deg, C);
            NTT(A, L, 1); NTT(C, L, 1);
            for (int i = 0; i < L; ++i) C[i] = (ll)A[i] * C[i] % P;
            NTT(C, L, -1);
            for (int i = 0; i < L; ++i) B[i] = (ll)(B[i] + C[i]) * Inv2 % P;
            for (int i = deg; i < L; ++i) B[i] = 0;
        }
        for (int i = 0; i < n; ++i) b[i] = B[i];
    }
}
using DFT::GetInv;
using DFT::GetSqrt;

int main() {
    #ifdef hzhkk
    freopen("hkk.in", "r", stdin);
    #endif
    read(n), read(m);
    for (int i = 1; i <= n; ++i) read(x), a[x] = 1;
    for (int i = 1; i <= m; ++i) if (a[i]) a[i] = P - SMOD(SMOD(a[i] << 1) << 1);
    a[0] = 1;
    GetSqrt(a, m + 1, a);  SADD(a[0], 1);
    GetInv(a, m + 1, a);
    for (int i = 1; i <= m; ++i) printf("%d\n", SMOD(a[i] << 1));
}