【LOJ6569】仙人掌计数

版权声明：本文为博主原创文章，未经博主允许不得转载。 https://blog.csdn.net/qq_39972971/article/details/89214832

【题目链接】

• 点击打开链接

【思路要点】

• 考虑对有根仙人掌的圆方树计数，定义子树大小为子树内圆点的个数。
• 令子树大小为 $i$ 的圆点和方点各有 $r_i,s_i$ 个，则其指数型生成函数分别为
  $R(x)=\sum_{i=1}\frac{r_i}{i!}x^i,S(x)=\sum_{i=1}\frac{s_i}{i!}x^i$
• 稍加推导可得
  $R(x)=xe^{S(x)},S(x)=R(x)+\frac{R(x)^2}{2}+\frac{R(x)^3}{2}+\dots=R(x)+\frac{R(x)^2}{2-2R(x)}$
• 因此有
  $R(x)=xe^{R(x)+\frac{R(x)^2}{2-2R(x)}}$
  $xe^{R(x)+\frac{R(x)^2}{2-2R(x)}}-R(x)=0$
• 令 $G(R(x))=xe^{R(x)+\frac{R(x)^2}{2-2R(x)}}-R(x)$ ，则 $G(R(x))=0$
• 考虑牛顿迭代解上述方程，令
  $R(x)\equiv R_0(x)\quad(mod\ x^n)$
• 则有 $G(R(x))\equiv G(R_0(x))+(R(x)-R_0(x))G'(R_0(x))\equiv0\quad(mod\ x^{2n})$
• 因此 $R(x)\equiv R_0(x)-\frac{G(R_0(x))}{G'(R_0(x))}\quad(mod\ x^{2n})$
• 其中 $G'(R(x))=\frac{dG(R(x))}{dR(x)}=xe^{R(x)+\frac{R(x)^2}{2-2R(x)}}(1+\frac{4R(x)-2R(x)^2}{(2-2R(x))^2})-1$
• 按此迭代，时间复杂度 $O(NLogN)$ 。
• 注意到我们计算的是有根仙人掌的个数，因此在最后应当将答案除去 $N$ 。

【代码】

#include<bits/stdc++.h>
using namespace std;
const int MAXN = 262144;
const int P = 998244353;
typedef long long ll;
typedef long double ld;
typedef unsigned long long ull;
template <typename T> void chkmax(T &x, T y) {x = max(x, y); }
template <typename T> void chkmin(T &x, T y) {x = min(x, y); } 
template <typename T> void read(T &x) {
	x = 0; int f = 1;
	char c = getchar();
	for (; !isdigit(c); c = getchar()) if (c == '-') f = -f;
	for (; isdigit(c); c = getchar()) x = x * 10 + c - '0';
	x *= f;
}
template <typename T> void write(T x) {
	if (x < 0) x = -x, putchar('-');
	if (x > 9) write(x / 10);
	putchar(x % 10 + '0');
}
template <typename T> void writeln(T x) {
	write(x);
	puts("");
}
namespace Poly {
	const int MAXN = 262144;
	const int P = 998244353;
	const int LOG = 25;
	const int G = 3;
	int power(int x, int y) {
		if (y == 0) return 1;
		int tmp = power(x, y / 2);
		if (y % 2 == 0) return 1ll * tmp * tmp % P;
		else return 1ll * tmp * tmp % P * x % P;
	}
	int invn[MAXN], tmpa[MAXN], tmpb[MAXN];
	int N, Log, home[MAXN]; bool initialized;
	int forward[MAXN], bckward[MAXN], inv[LOG];
	void init() {
		initialized = true;
		forward[0] = bckward[0] = inv[0] = invn[1] = 1;
		for (int len = 2, lg = 1; len <= MAXN; len <<= 1, lg++)
			inv[lg] = power(len, P - 2);
		for (int i = 2; i < MAXN; i++)
			invn[i] = (P - 1ll * (P / i) * invn[P % i] % P) % P;
		int delta = power(G, (P - 1) / MAXN);
		for (int i = 1; i < MAXN; i++)
			forward[i] = bckward[MAXN - i] = 1ll * forward[i - 1] * delta % P;
	}
	void NTTinit() {
		for (int i = 0; i < N; i++) {
			int ans = 0, tmp = i;
			for (int j = 1; j <= Log; j++) {
				ans <<= 1;
				ans += tmp & 1;
				tmp >>= 1;
			}
			home[i] = ans;
		}
	}
	void NTT(int *a, int mode) {
		assert(initialized);
		for (int i = 0; i < N; i++)
			if (home[i] < i) swap(a[i], a[home[i]]);
		int *g;
		if (mode == 1) g = forward;
		else g = bckward;
		for (int len = 2, lg = 1; len <= N; len <<= 1, lg++) {
			for (int i = 0; i < N; i += len) {
				for (int j = i, k = i + len / 2; k < i + len; j++, k++) {
					int tmp = a[j];
					int tnp = 1ll * a[k] * g[MAXN / len * (j - i)] % P;
					a[j] = (tmp + tnp > P) ? (tmp + tnp - P) : (tmp + tnp);
					a[k] = (tmp - tnp < 0) ? (tmp - tnp + P) : (tmp - tnp);
				}
			}
		}
		if (mode == -1) {
			for (int i = 0; i < N; i++)
				a[i] = 1ll * a[i] * inv[Log] % P;
		}
	}
	void times(vector <int> &a, vector <int> &b, vector <int> &c) {
		assert(a.size() >= 1), assert(b.size() >= 1);
		int goal = a.size() + b.size() - 1;
		N = 1, Log = 0;
		while (N < goal) {
			N <<= 1;
			Log++;
		}
		for (unsigned i = 0; i < a.size(); i++)
			tmpa[i] = a[i];
		for (int i = a.size(); i < N; i++)
			tmpa[i] = 0;
		for (unsigned i = 0; i < b.size(); i++)
			tmpb[i] = b[i];
		for (int i = b.size(); i < N; i++)
			tmpb[i] = 0;
		NTTinit();
		NTT(tmpa, 1);
		NTT(tmpb, 1);
		for (int i = 0; i < N; i++)
			tmpa[i] = 1ll * tmpa[i] * tmpb[i] % P;
		NTT(tmpa, -1);
		c.resize(goal);
		for (int i = 0; i < goal; i++)
			c[i] = tmpa[i];
	}
	void timesabb(vector <int> &a, vector <int> &b, vector <int> &c) {
		assert(a.size() >= 1), assert(b.size() >= 1);
		int goal = a.size() + b.size() * 2 - 2;
		N = 1, Log = 0;
		while (N < goal) {
			N <<= 1;
			Log++;
		}
		for (unsigned i = 0; i < a.size(); i++)
			tmpa[i] = a[i];
		for (int i = a.size(); i < N; i++)
			tmpa[i] = 0;
		for (unsigned i = 0; i < b.size(); i++)
			tmpb[i] = b[i];
		for (int i = b.size(); i < N; i++)
			tmpb[i] = 0;
		NTTinit();
		NTT(tmpa, 1);
		NTT(tmpb, 1);
		for (int i = 0; i < N; i++)
			tmpa[i] = 1ll * tmpa[i] * tmpb[i] % P * tmpb[i] % P;
		NTT(tmpa, -1);
		c.resize(goal);
		for (int i = 0; i < goal; i++)
			c[i] = tmpa[i];
	}
	void getinv(vector <int> &a, vector <int> &b) {
		assert(a.size() >= 1), assert(a[0] != 0);
		b.clear(), b.push_back(power(a[0], P - 2));
		while (b.size() < a.size()) {
			vector <int> c, ta = a;
			ta.resize(b.size() * 2);
			timesabb(ta, b, c);
			b.resize(b.size() * 2);
			for (unsigned i = 0; i < b.size(); i++)
				b[i] = (2ll * b[i] - c[i] + P) % P;
		}
		b.resize(a.size());
	}
	void getder(vector <int> &a, vector <int> &b) {
		assert(a.size() >= 1);
		if (a.size() == 1) {
			b.clear();
			b.resize(1);
		} else {
			b.resize(a.size() - 1);
			for (unsigned i = 0; i < b.size(); i++)
				b[i] = (i + 1ll) * a[i + 1] % P;
		}
	}
	void getint(vector <int> &a, vector <int> &b) {
		b.resize(a.size() + 1), b[0] = 0;
		for (unsigned i = 0; i < a.size(); i++)
			b[i + 1] = 1ll * invn[i + 1] * a[i] % P;
	}
	void getlog(vector <int> &a, vector <int> &b) {
		assert(a.size() >= 1), assert(a[0] == 1);
		vector <int> da, inva, db;
		getder(a, da), getinv(a, inva);
		times(da, inva, db), getint(db, b);
		b.resize(a.size());
	}
	void getexp(vector <int> &a, vector <int> &b) {
		assert(a.size() >= 1), assert(a[0] == 0);
		b.clear(), b.push_back(1);
		while (b.size() < a.size()) {
			vector <int> lnb, res;
			b.resize(b.size() * 2), getlog(b, lnb);
			for (unsigned i = 0; i < lnb.size(); i++)
				if (i == 0) lnb[i] = (P + 1 + a[i] - lnb[i]) % P;
				else if (i < a.size()) lnb[i] = (P + a[i] - lnb[i]) % P;
				else lnb[i] = (P - lnb[i]) % P;
			times(lnb, b, res);
			res.resize(b.size());
			swap(res, b);
		}
		b.resize(a.size());
	}
	void getshl(vector <int> &a, vector <int> &b, ull bits) {
		if (a.size() < bits) bits = a.size();
		b.clear(), b.resize(bits);
		for (unsigned i = 0; b.size() < a.size(); i++)
			b.push_back(a[i]);
	}
	void getshr(vector <int> &a, vector <int> &b, ull bits) {
		if (a.size() < bits) bits = a.size(); b.clear();
		for (unsigned i = bits; i < a.size(); i++)
			b.push_back(a[i]);
		b.resize(a.size());
	}
	void getpowk(vector <int> &a, vector <int> &b, int k) {
		assert(k >= 1);
		unsigned pos = a.size();
		for (unsigned i = 0; i < a.size(); i++)
			if (a[i]) {
				pos = i;
				break;
			}
		if (pos == a.size()) {
			b = a;
			return;
		}
		int val = power(a[pos], k), inv = power(a[pos], P - 2);
		vector <int> lntmp, tmp;
		getshr(a, tmp, pos);
		for (unsigned i = 0; i < tmp.size(); i++)
			tmp[i] = 1ll * tmp[i] * inv % P;
		getlog(tmp, lntmp);
		for (unsigned i = 0; i < lntmp.size(); i++)
			lntmp[i] = 1ll * lntmp[i] * k % P;
		getexp(lntmp, tmp);
		for (unsigned i = 0; i < tmp.size(); i++)
			tmp[i] = 1ll * tmp[i] * val % P;
		getshl(tmp, b, 1ull * pos * k);
	}
	int getinvfunc(vector <int> &f, unsigned n) {
		assert(f[0] == 0 && f[1] != 0);
		assert(n >= 1 && n <= f.size());
		int inv = power(n, P - 2);
		vector <int> tmp;
		getshr(f, tmp, 1);
		vector <int> invf;
		getinv(tmp, invf);
		vector <int> res;
		getpowk(invf, res, n);
		return 1ll * inv * res[n - 1] % P;
	}
}
int fac[MAXN], inv[MAXN];
int power(int x, int y) {
	if (y == 0) return 1;
	int tmp = power(x, y / 2);
	if (y % 2 == 0) return 1ll * tmp * tmp % P;
	else return 1ll * tmp * tmp % P * x % P;
}
void init(int n) {
	Poly :: init();
	fac[0] = 1;
	for (int i = 1; i <= n; i++)
		fac[i] = 1ll * fac[i - 1] * i % P;
	inv[n] = power(fac[n], P - 2);
	for (int i = n - 1; i >= 0; i--)
		inv[i] = inv[i + 1] * (i + 1ll) % P;
}
void update(int &x, int y) {
	x += y;
	if (x >= P) x -= P;
}
int main() {
	init(131072);
	vector <int> r;
	r.push_back(0);
	r.push_back(1);
	while (r.size() < 131072) {
		vector <int> d, f, invf, l, e;
		Poly :: times(r, r, d);
		r.resize(r.size() * 2);
		d.resize(r.size());
		f.resize(r.size());
		for (unsigned i = 0; i < r.size(); i++) {
			d[i] = (2ll * r[i] - d[i] + P) % P;
			f[i] = 1ll * (P - 2) * r[i] % P;
			if (i == 0) update(f[i], 2);
		}
		Poly :: getinv(f, invf);
		Poly :: times(d, invf, l);
		l.resize(r.size());
		Poly :: getexp(l, e);
		vector <int> g, dg, invdg, res;
		Poly :: getshl(e, g, 1);
		for (unsigned i = 0; i < r.size(); i++)
			d[i] = 2ll * d[i] % P;
		Poly :: times(invf, invf, f);
		f.resize(r.size());
		Poly :: times(d, f, e);
		e.resize(r.size());
		update(e[0], 1);
		Poly :: times(g, e, dg);
		dg.resize(r.size());
		update(dg[0], P - 1);
		Poly :: getinv(dg, invdg);
		for (unsigned i = 0; i < r.size(); i++)
			update(g[i], P - r[i]);
		Poly :: times(g, invdg, res);
		res.resize(r.size());
		for (unsigned i = 0; i < r.size(); i++)
			update(r[i], P - res[i]);
	}
	int T; read(T);
	while (T--) {
		int x; read(x);
		writeln(1ll * fac[x - 1] * r[x] % P);
	}
	return 0;
}