cf438E. The Child and Binary Tree(生成函数 多项式开根 多项式求逆)

题意

链接

sol

生成函数博大精深orz

我们设\(f(i)\)表示权值为\(i\)的二叉树数量，转移的时候可以枚举一下根节点

\(f(n) = \sum_{w \in c_1 \dots c_n} \sum_{j=0}^{n-w} f(j) f(n-w-j)\)

设\(t =n-w\)，后半部分变为\(\sum_{j=0}^t f(j) f(t-j)\)，是个标准的卷积形式。

对于第一重循环我们可以设出现过的数的生成函数\(c(x)\)

可以得到\(f = c * f * f + 1\)，+1是因为\(f[0] = 1\)

可以解得\(f = \frac{1\pm\sqrt{1-4g}}{2g} = \frac{2}{1\pm\sqrt{1-4c}}\)

现在问题来了，我们是要取\(+\)还是取\(-\)。

结论是取\(+\)，因为当取\(-\)时，c中x的取值趋向于\(0\)时分母会无意义

举个例子(来自cf讨论区)

\(c = 2x - 4x^2\)，\(+\sqrt{1-4c} = 1 - 4x, -\sqrt{1-4c} = -1+4x\)

后者带入得到\(f = \frac{2}{4x}\)，这玩意儿显然是无解的，因为多项式有逆元的充要条件是常数项在模意义下有逆元，然而这玩意儿的常数项是0.。

感觉做这种题直接还是要先推一推暴力dp的式子吧，不然直接用生成函数推根本无从下手。。

#include<bits/stdc++.h> 
#define pair pair<int, int>
#define mp(x, y) make_pair(x, y)
#define fi first
#define se second
#define ll long long 
#define ull unsigned long long 
#define fin(x) {freopen(#x".in","r",stdin);}
#define fout(x) {freopen(#x".out","w",stdout);}
using namespace std;
const int maxn = 1e6 + 10, inf = 1e9 + 1;
const double eps = 1e-9, pi = acos(-1);
inline int read() {
    char c = getchar(); int x = 0, f = 1;
    while(c < '0' || c > '9') {if(c == '-') f = -1; c = getchar();}
    while(c >= '0' && c <= '9') x = x * 10 + c - '0', c = getchar();
    return x * f;
}
int n, m, a[maxn], b[maxn], c[maxn], d[maxn];
namespace poly {
    int rev[maxn], gpow[maxn], a[maxn], b[maxn], c[maxn], lim, inv2;
    const int g = 3, mod = 998244353;
    template <typename a, typename b> inline ll add(a x, b y) {if(x + y < 0) return x + y + mod; return x + y >= mod ? x + y - mod : x + y;}
    template <typename a, typename b> inline void add2(a &x, b y) {if(x + y < 0) x = x + y + mod; else x = (x + y >= mod ? x + y - mod : x + y);}
    template <typename a, typename b> inline ll mul(a x, b y) {return 1ll * x * y % mod;}
    template <typename a, typename b> inline void mul2(a &x, b y) {x = (1ll * x * y % mod + mod) % mod;}
    int fp(int a, int p, int p = mod) {
        int base = 1;
        for(; p; p >>= 1, a = 1ll * a * a % p) if(p & 1) base = 1ll * base *  a % p;
        return base;
    }
    int getlen(int x) {
        int lim = 1;
        while(lim <= x) lim <<= 1;
        return lim;
    }
    int getorigin(int x) {//¼æëãô­¸ù 
        static int q[maxn]; int tot = 0, tp = x - 1;
        for(int i = 2; i * i <= tp; i++) if(!(tp % i)) {q[++tot] = i;while(!(tp % i)) tp /= i;}
        if(tp > 1) q[++tot] = tp;
        for(int i = 2, j; i <= x - 1; i++) {
            for(j = 1; j <= tot; j++) if(fp(i, (x - 1) / q[j], x) == 1) break;
            if(j == tot + 1) return i;
        }
    }
    void init(/*int p,*/ int lim) {
        //mod = p; g = getorigin(mod); gi = fp(g, mod - 2);
        inv2 = fp(2, mod - 2);
        for(int i = 1; i < lim; i++) gpow[i] = fp(g, (mod - 1) / i);
    }
    void ntt(int *a, int lim, int opt) {
        int len = 0; for(int n = 1; n < lim; n <<= 1) ++len; 
        for(int i = 1; i <= lim; i++) rev[i] = (rev[i >> 1] >> 1) | ((i & 1) << (len - 1));
        for(int i = 0; i <= lim; i++) if(i < rev[i]) swap(a[i], a[rev[i]]);
        for(int mid = 1; mid < lim; mid <<= 1) {
            int wn = gpow[mid << 1];
            for(int i = 0; i < lim; i += (mid << 1)) {
                for(int j = 0, w = 1; j < mid; j++, w = mul(w, wn)) {
                    int x = a[i + j], y = mul(w, a[i + j + mid]);
                    a[i + j] = add(x, y), a[i + j + mid] = add(x, -y);
                }
            }
        }
        if(opt == -1) {
            reverse(a + 1, a + lim);
            int inv = fp(lim, mod - 2);
            for(int i = 0; i <= lim; i++) mul2(a[i], inv);
        }
    }
    void mul(int *a, int *b, int n, int m) {
        memset(a, 0, sizeof(a)); memset(b, 0, sizeof(b));
        int lim = 1, len = 0; 
        while(lim <= n + m) len++, lim <<= 1;
        for(int i = 0; i <= n; i++) a[i] = a[i]; 
        for(int i = 0; i <= m; i++) b[i] = b[i];
        ntt(a, lim, 1); ntt(b, lim, 1);
        for(int i = 0; i <= lim; i++) b[i] = mul(b[i], a[i]);
        ntt(b, lim, -1);
        for(int i = 0; i <= n + m; i++) b[i] = b[i];
        memset(a, 0, sizeof(a)); memset(b, 0, sizeof(b));
    }
    void inv(int *a, int *b, int len) {//b1 = 2b - a1 * b^2 
        if(len == 1) {b[0] = fp(a[0], mod - 2); return ;}
        inv(a, b, len >> 1);
        for(int i = 0; i < len; i++) a[i] = a[i], b[i] = b[i];
        ntt(a, len << 1, 1); ntt(b, len << 1, 1);
        for(int i = 0; i < (len << 1); i++) mul2(a[i], mul(b[i], b[i]));
        ntt(a, len << 1, -1);
        for(int i = 0; i < len; i++) add2(b[i], add(b[i], -a[i]));
        for(int i = 0; i < (len << 1); i++) a[i] = b[i] = 0;
    }
    void dao(int *a, int *b, int len) {
        for(int i = 1; i < len; i++) b[i - 1] = mul(i, a[i]); b[len - 1] = 0;
    }
    void ji(int *a, int *b, int len) {
        for(int i = 1; i < len; i++) b[i] = mul(a[i - 1], fp(i, mod - 2)); b[0] = 0;
    }
    void ln(int *a, int *b, int len) {//g(a) = \frac{a}{a'} qiudao zhihou jifen 
        static int a[maxn], b[maxn];
        dao(a, a, len); 
        inv(a, b, len);
        ntt(a, len << 1, 1); ntt(b, len << 1, 1);
        for(int i = 0; i < (len << 1); i++) b[i] = mul(a[i], b[i]);
        ntt(b, len << 1, -1); 
        ji(b, b, len << 1);
        memset(a, 0, sizeof(a)); memset(b, 0, sizeof(b));
    }
    void exp(int *a, int *b, int len) {//f(x) = f_0 (1 - lnf_0 + a) but code ..why....
        if(len == 1) return (void) (b[0] = 1);
        exp(a, b, len >> 1); ln(b, c, len);
        c[0] = add(a[0] + 1, -c[0]);
        for(int i = 1; i < len; i++) c[i] = add(a[i], -c[i]);
        ntt(c, len << 1, 1); ntt(b, len << 1, 1);
        for(int i = 0; i < (len << 1); i++) mul2(b[i], c[i]);
        ntt(b, len << 1, -1);
        for(int i = len; i < (len << 1); i++) c[i] = b[i] = 0;
    }
    void sqrt(int *a, int *b, int len) {
        static int b[maxn];
        ln(a, b, len);
        for(int i = 0; i < len; i++) b[i] = mul(b[i], inv2);
        exp(b, b, len); 
    }
};
using namespace poly; 

signed main() {
    n = read(); m = read(); int lim = getlen(m); init(4 * lim);
    for(int i = 1; i <= n; i++) a[i] = read();
    for(int i = 1; i <= n; i++) b[a[i]] = (-4 + mod); add2(b[0], 1);
    sqrt(b, c, lim);
    add2(c[0], 1);
    inv(c, d, lim);
    for(int i = 1; i <= m; i++) cout << mul(2, d[i]) << '\n';
    return 0;
}