luoguP4705 玩游戏 分治FFT

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\[\begin{aligned} Ans(k) &= \sum \limits_{i = 1}^n \sum \limits_{j = 1}^m \sum \limits_{t = 0}^k \binom{k}{t} a_i^t b_j^{k - t} \\ &= \sum \limits_{t = 0}^k \binom{k}{t} (\sum \limits_{i = 1}^n a_i^t) (\sum \limits_{j = 1}^m b_i^{k - t}) \\ &= k! * \sum \limits_{t = 0}^k (\frac{\sum \limits_{i = 1}^n a_i^t}{t!}) (\frac{\sum \limits_{j = 1}^m b_i^{k - t}}{(k - t)!}) \\ \end{aligned} \]

右边是一个卷积，只需考虑对\(t = 0, 1 ..., n\)求出\(f(t) = \sum \limits_{i = 1}^n a_i^t\)

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考虑生成函数\(OGF\)

\[\begin{aligned} F(x) &= \sum \limits_{i = 0}^{\infty} f(i) * x^i \\ &= \sum \limits_{i = 1}^{\infty} \sum \limits_{j = 1}^{n} a_j^i \\ &= \sum \limits_{i = 1}^n (\sum \limits_{j = 0}^{\infty} (a_ix)^j) \\ &= \sum \limits_{i = 1}^n \frac{1}{1 - a_i x} \end{aligned} \]

那么，现在的问题在于如何求解

\[\sum \limits_{i = 1}^n \frac{1}{1 - a_i x} \]

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考虑分治\(FFT\)
一种很好想的思路是先求出\(\sum \limits_{i = 1}^l \frac{1}{1 - a_i x}\)和\(\sum \limits_{i = l + 1}^r \frac{1}{1 - a_i x}\)
它们一定是形如\(\frac{A}{B}\)的一个式子，不妨设左边为\(\frac{A}{B}\)，右边为\(\frac{C}{D}\)
那么合并之后的形式为\(\frac{AD + BC}{BD}\)，然后维护即可

复杂度\(O(n log^2 n)\)

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可以发现$$In'(a * b) = (In(a) + In(b))' = In'(a) + In'(b)$$
因此，我们考虑$$In'(\frac{1}{1 - a_i x}) = \frac{-a_i}{1 - a_i x}$$
注意不能在\(In\)中添加常数因子，因此我们只能从这个形式来考虑
令

\[\begin{aligned} G(x) &= \sum \limits_{i = 1}^n In'(\frac{1}{1 - a_i x}) \\ &= In ' (\prod \limits_{i = 1}^n \frac{1}{1 - a_i x}) \end{aligned} \]

可以用分治\(FFT\)求出\(G\)
观察数列

\[F(x) = a_i^0 + a_i^1 x^1 + a_i^2 x^2 + a_i^3 x^3 ... \]

\[G(x) = -a_i^1 - a_i^2 x^1 - a_i^3 x^2 + ... \]

因此，$$-xG(x) + n = F(x)$$

然后就可以啦

---

#include <bits/stdc++.h>
using namespace std;

#define ri register int
#define rep(io, st, ed) for(ri io = st; io <= ed; io ++)
#define drep(io, ed, st) for(ri io = ed; io >= st; io --)

const int sid = 5e5 + 5;
const int mod = 998244353;

#define gc getchar
inline int read() {
	int p = 0, w = 1; char c = gc();
	while(c > '9' || c < '0') { if(c == '-') w = -1; c = gc(); }
	while(c >= '0' && c <= '9') p = p * 10 + c - '0', c = gc();
	return p * w;
}

inline int Inc(int a, int b) { return (a + b >= mod) ? a + b - mod : a + b; }
inline int Dec(int a, int b) { return (a - b < 0) ? a - b + mod : a - b; }
inline int mul(int a, int b) { return 1ll * a * b % mod; }
inline int fp(int a, int k) {
	int ret = 1;
	for( ; k; k >>= 1, a = mul(a, a))
		if(k & 1) ret = mul(ret, a);
	return ret;
}
	
int rev[sid], fac[sid], inv[sid], ivf[sid];
int a[sid], b[sid], ak[sid], bk[sid];
	
inline void init(int Mn, int &n, int &lg) {
	n = 1; lg = 0;
	while(n < Mn) n <<= 1, lg ++;
}
	
inline void NTT(int *a, int n, int opt) {
	for(ri i = 0; i < n; i ++) 
		if(i < rev[i]) swap(a[i], a[rev[i]]);
	for(ri i = 1; i < n; i <<= 1)
	for(ri j = 0, g = fp(3, (mod - 1) / (i << 1)); j < n; j += (i << 1))
	for(ri k = j, G = 1; k < i + j; k ++, G = mul(G, g)) {
		int x = a[k], y = mul(G, a[i + k]);
		a[k] = (x + y >= mod) ? x + y - mod : x + y;
		a[i + k] = (x - y < 0) ? x - y + mod : x - y;
	}
	if(opt == -1) {
		reverse(a + 1, a + n);
		int ivn = fp(n, mod - 2);
		for(ri i = 0; i < n; i ++) a[i] = mul(a[i], ivn);
	}
}
	
int ia[sid], ib[sid];
inline void Inv(int *a, int *b, int n) {
	if(n == 1) { b[0] = fp(a[0], mod - 2); return; }
	Inv(a, b, n >> 1);
	
	int N = 1, lg = 0; init(n + n, N, lg);
	for(ri i = 0; i < N; i ++) rev[i] = (rev[i >> 1] >> 1) | ((i & 1) << (lg - 1));
	for(ri i = 0; i < N; i ++) ia[i] = ib[i] = 0;
	for(ri i = 0; i < n; i ++) ia[i] = a[i], ib[i] = b[i];
	
	NTT(ia, N, 1); NTT(ib, N, 1);
	for(ri i = 0; i < N; i ++) ia[i] = Dec((ib[i] << 1) % mod, mul(ia[i], mul(ib[i], ib[i])));
	NTT(ia, N, -1);
	
	for(ri i = 0; i < n; i ++) b[i] = ia[i];
}
	
inline void Init_Inv(int n) {
	inv[0] = inv[1] = 1;
	for(int i = 2; i <= n; i ++) inv[i] = mul(inv[mod % i], mod - mod / i);
	fac[0] = fac[1] = 1;
	for(int i = 2; i <= n; i ++) fac[i] = mul(fac[i - 1], i);
	ivf[0] = ivf[1] = 1;
	for(int i = 2; i <= n; i ++) ivf[i] = mul(ivf[i - 1], inv[i]);
}
	
inline void wf(int *a, int *b, int n) { for(ri i = 1; i < n; i ++) b[i - 1] = mul(a[i], i); }
inline void jf(int *a, int *b, int n) { for(ri i = 1; i < n; i ++) b[i] = mul(a[i - 1], inv[i]);}
	
int da[sid], iva[sid];
inline void In(int *a, int *b, int n) {
	for(ri i = 0; i < n + n; i ++) da[i] = iva[i] = 0; 
	Inv(a, iva, n); wf(a, da, n);
	
	int N = 1, lg = 0; init(n + n, N, lg);
	for(ri i = 0; i < N; i ++) rev[i] = (rev[i >> 1] >> 1) | ((i & 1) << (lg - 1));
	
	NTT(da, N, 1); NTT(iva, N, 1);
	for(ri i = 0; i < N; i ++) da[i] = mul(da[i], iva[i]);
	NTT(da, N, -1); jf(da, b, n);
}

int hb[sid], inb[sid];
inline void Exp(int *a, int *b, int n) {
	if(n == 1) { b[0] = 1; return; }
	Exp(a, b, n >> 1);
	
	int N = 1, lg = 0; init(n + n, N, lg);
	for(ri i = 0; i < N; i ++) rev[i] = (rev[i >> 1] >> 1) | ((i & 1) << (lg - 1));
	
	for(ri i = 0; i < N; i ++) inb[i] = hb[i] = 0;
	In(b, inb, n);
	for(ri i = 0; i < n; i ++) hb[i] = Dec(a[i], inb[i]); hb[0] ++;
	
	NTT(inb, N, 1); NTT(hb, N, 1);
	for(ri i = 0; i < N; i ++) inb[i] = mul(inb[i], hb[i]);
	NTT(inb, N, -1);
	
	for(ri i = 0; i < n; i ++) b[i] = inb[i];
}

int Ib[sid], F[sid * 2], pa[sid], pb[sid];
inline void calc(int *a, int *b, int n, int t) {
	int N = 1, lg = 0;
	init(max(n, t) + 5, N, lg);
	for(ri i = 0; i < (N << 1); i ++) F[i] = 0;
	for(ri i = 0; i < n; i ++) F[2 * i] = 1, F[2 * i + 1] = mod - a[i + 1];
	for(ri i = n; i < N; i ++) F[2 * i] = 1;
	for(ri i = 1; i < N; i <<= 1) {
		for(ri j = 0; j < N; j += (i << 1)) {
			int M = 1, lg = 0;
			init((i << 2), M, lg);
			for(ri k = 0; k < M; k ++) rev[k] = (rev[k >> 1] >> 1) | ((k & 1) << (lg - 1));
			for(ri k = 0; k < M; k ++) pa[k] = pb[k] = 0;
			for(ri k = 0; k < (i << 1); k ++) 
				pa[k] = F[(j << 1) + k], pb[k] = F[(j << 1) + (i << 1) + k];
			NTT(pa, M, 1); NTT(pb, M, 1);
			for(ri k = 0; k < M; k ++) pa[k] = mul(pa[k], pb[k]);
			NTT(pa, M, -1);
			for(ri k = 0; k < (i << 2); k ++) F[(j << 1) + k] = pa[k];
		}
	}
	for(ri i = 0; i < N; i ++) Ib[i] = 0;
	In(F, Ib, N); wf(Ib, F, N);
	
	b[0] = n;
	for(ri i = 1; i <= t; i ++) b[i] = mul(mod - F[i - 1], ivf[i]);
}
	
inline void solve(int n, int m, int t) {
	int N = 1, lg = 0;
	init(t + t + 5, N, lg);
	for(ri i = 0; i < N; i ++) rev[i] = (rev[i >> 1] >> 1) | ((i & 1) << (lg - 1));
	
	NTT(ak, N, 1); NTT(bk, N, 1);
	for(ri i = 0; i < N; i ++) ak[i] = mul(ak[i], bk[i]);
	NTT(ak, N, -1);
		
	int ivnm = fp(mul(n, m), mod - 2);
	for(ri i = 1; i <= t; i ++) 
		printf("%d\n", mul(mul(ak[i], fac[i]), ivnm));
}
	
int main() {
	int n = read(), m = read();
	rep(i, 1, n) a[i] = read();
	rep(i, 1, m) b[i] = read();
	Init_Inv(500000);
	int t = read(); 
	calc(a, ak, n, t); calc(b, bk, m, t);
	solve(n, m, t);
	return 0;
}

请无视中间的exp