题解 「THUPC 2017」小 L 的计算题 / Sum

题目传送门

题目大意

给出 \(a_{1,2,...,n}\)，对于 \(\forall k\in [1,n]\) ，求出：

\[\sum_{i=1}^{n}a_i^k
\]

\(n\le 2\times 10^5\)，答案对 \(998244353\) 取模 。

思路

我们考虑对答案构造生成函数：

\[F(x)=\sum_{k=0}^{\infty} \sum_{i=1}^{n}a_i^kx^k
\]

\[=\sum_{i=1}^{n}\frac{1}{1-a_ix}
\]

\[=\sum_{i=1}^{n}(1+\frac{a_ix}{1-a_ix})
\]

\[=n-x(\sum_{i=1}^{n}\frac{-a_i}{1-a_ix})
\]

然后我们发现存在：

\[(\ln(1-a_ix))^{'}=\frac{-a_i}{1-a_ix}
\]

于是：

\[F(x)=n-x(\sum_{i=1}^{n}\ln(1-a_ix))^{'}
\]

\[=n-x(\ln \prod_{i=1}^{n}(1-a_ix))^{'}
\]

然后这个东西就可以直接分治解决了，时间复杂度 \(\Theta(n\log^2 n)\) 。

\(\texttt{Code}\)

#pragma GCC optimize("Ofast")
#pragma GCC optimize("inline", "no-stack-protector", "unroll-loops")
#pragma GCC diagnostic error "-fwhole-program"
#pragma GCC diagnostic error "-fcse-skip-blocks"
#pragma GCC diagnostic error "-funsafe-loop-optimizations"

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

#define SZ(x) ((int)x.size())
#define Int register int
#define mod 998244353
#define MAXN 1000005

int mul (int a,int b){return 1ll * a * b % mod;}
int dec (int a,int b){return a >= b ? a - b : a + mod - b;}
int add (int a,int b){return a + b >= mod ? a + b - mod : a + b;}
int qkpow (int a,int k){
	int res = 1;for (;k;k >>= 1,a = 1ll * a * a % mod) if (k & 1) res = 1ll * res * a % mod;
	return res;
}
int inv (int x){return qkpow (x,mod - 2);}

typedef vector <int> poly;

int w[MAXN],rev[MAXN];

void init_ntt (){
	int lim = 1 << 19;
	for (Int i = 0;i < lim;++ i) rev[i] = (rev[i >> 1] >> 1) | ((i & 1) << 18);
	int Wn = qkpow (3,(mod - 1) / lim);w[lim >> 1] = 1;
	for (Int i = lim / 2 + 1;i < lim;++ i) w[i] = mul (w[i - 1],Wn);
	for (Int i = lim / 2 - 1;i;-- i) w[i] = w[i << 1];
}

void ntt (poly &a,int lim,int type){
#define G 3
#define Gi 332748118
	static unsigned long long d[MAXN];
	for (Int i = 0,z = 19 - __builtin_ctz(lim);i < lim;++ i) d[rev[i] >> z] = a[i];
	for (Int i = 1;i < lim;i <<= 1)
		for (Int j = 0;j < lim;j += i << 1)
			for (Int k = 0;k < i;++ k){
				int x = mul (w[i + k],d[i + j + k]);
				d[i + j + k] = dec (d[j + k],x),d[j + k] = add (d[j + k],x);
			}
	for (Int i = 0;i < lim;++ i) a[i] = d[i];
	if (type == -1){
		reverse (a.begin() + 1,a.begin() + lim);
		for (Int i = 0,Inv = inv (lim);i < lim;++ i) a[i] = mul (a[i],Inv);
	}
#undef G
#undef Gi
}

poly operator * (poly a,poly b){
	int d = SZ (a) + SZ (b) - 1,lim = 1;while (lim < d) lim <<= 1;
	a.resize (lim),b.resize (lim);
	ntt (a,lim,1),ntt (b,lim,1);
	for (Int i = 0;i < lim;++ i) a[i] = mul (a[i],b[i]);
	ntt (a,lim,-1),a.resize (d);
	return a;
}

poly inv (poly a,int n){
	poly b(1,inv (a[0])),c;
	for (Int l = 4;(l >> 2) < n;l <<= 1){
		c.resize (l >> 1);
		for (Int i = 0;i < (l >> 1);++ i) c[i] = i < n ? a[i] : 0;
		c.resize (l),b.resize (l);
		ntt (c,l,1),ntt (b,l,1);
		for (Int i = 0;i < l;++ i) b[i] = mul (b[i],dec (2,mul (b[i],c[i])));
		ntt (b,l,-1),b.resize (l >> 1);
	}
	b.resize (n);
	return b;
}

poly inv (poly a){return inv (a,SZ (a));}

poly der (poly a){
	for (Int i = 0;i < SZ (a) - 1;++ i) a[i] = mul (a[i + 1],i + 1);
	a.pop_back ();return a;
}

poly ine (poly a){
	a.push_back (0);
	for (Int i = SZ (a) - 1;i;-- i) a[i] = mul (a[i - 1],inv (i));
	a[0] = 0;return a;
}

poly ln (poly a,int n){
	a = ine (der (a) * inv (a));
	a.resize (n);
	return a;
}

poly ln (poly a){return ln (a,SZ (a));}

template <typename T> inline void read (T &t){t = 0;char c = getchar();int f = 1;while (c < '0' || c > '9'){if (c == '-') f = -f;c = getchar();}while (c >= '0' && c <= '9'){t = (t << 3) + (t << 1) + c - '0';c = getchar();} t *= f;}
template <typename T,typename ... Args> inline void read (T &t,Args&... args){read (t);read (args...);}
template <typename T> inline void write (T x){if (x < 0){x = -x;putchar ('-');}if (x > 9) write (x / 10);putchar (x % 10 + '0');}

int n,a[MAXN];poly F;

poly divide (int l,int r){
	if (l == r){
		poly tmp;tmp.resize(2);
		tmp[0] = 1,tmp[1] = mod - a[l];
		return tmp;
 	}
	int mid = (l + r) >> 1;
	return divide (l,mid) * divide (mid + 1,r);
}

signed main(){
	init_ntt ();
	int T;read (T);
	while (T --> 0){
		read (n);
		for (Int i = 1;i <= n;++ i) read (a[i]);
		F = der (ln (divide (1,n))),F.resize (n + 1);
		for (Int i = n;i;-- i) F[i] = mod - F[i - 1];F[0] = n;
		int ans = 0;for (Int i = 1;i <= n;++ i) ans ^= (F[i] % mod + mod) % mod;
		write (ans),putchar ('\n');
	}
	return 0;
}