LOJ #6261. 一个人的高三楼 NTT

题目链接

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题意:求一个长度为\(n\)的数列的\(k\)阶前缀和对998244353取模的结果.\((n\le10^{5}\ \ k\le2^{60})\)

设$$f_k(x)=\sum_{i=0}^{n-1}S_{i+1}x^i$$

\[g(x)=\sum_{i=0}^{n-1}x^i \]

于是有$$f_k(x)g(x)\equiv f_{k+1}(x)\pmod{x^n}$$

所求即$$g(x)^kf_0(x)$$

直接快速幂 复杂度\(O(n\ log\ n\ log\ k)\) 无法通过此题

考虑\(g(x)^k\)各项的意义，设$$g(x)^k=\sum_{i=0}a_ix^i$$

\(a_i\)即为有序地选k个\([0,n-1]\)之间的整数，和为i的方案数，用插板法可得\(a_i=\binom{k-1+i}{i}\)，从而可以\(O(n)\)或\(O(n\ log\ n)\)得到\(g(x)^k\)

最后用\(NTT\)将\(f_0(x)\)和\(g(x)^k\)合并

总复杂度\(O(n\ log\ n)\)

#include<cstdio>
#include<algorithm>
#include<ctype.h>
#include<string.h>
#include<math.h>

using namespace std;
#define ll long long
#define rep(i,x,y) for(int i=(x);i<=(y);++i)
#define travel(i,x) for(int i=h[x];i;i=pre[i])

inline char read() {
	static const int IN_LEN = 1000000;
	static char buf[IN_LEN], *s, *t;
	return (s == t ? t = (s = buf) + fread(buf, 1, IN_LEN, stdin), (s == t ? -1 : *s++) : *s++);
}
template<class T>
inline void read(T &x) {
	static bool iosig;
	static char c;
	for (iosig = false, c = read(); !isdigit(c); c = read()) {
		if (c == '-') iosig = true;
		if (c == -1) return;
	}
	for (x = 0; isdigit(c); c = read()) x = ((x + (x << 2)) << 1) + (c ^ '0');
	if (iosig) x = -x;
}
const int OUT_LEN = 10000000;
char obuf[OUT_LEN], *ooh = obuf;
inline void print(char c) {
	if (ooh == obuf + OUT_LEN) fwrite(obuf, 1, OUT_LEN, stdout), ooh = obuf;
	*ooh++ = c;
}
template<class T>
inline void print(T x) {
	static int buf[30], cnt;
	if (x == 0) print('0');
	else {
		if (x < 0) print('-'), x = -x;
		for (cnt = 0; x; x /= 10) buf[++cnt] = x % 10 + 48;
		while (cnt) print((char)buf[cnt--]);
	}
}
inline void flush() { fwrite(obuf, 1, ooh - obuf, stdout); }
const int N = 1<<18, P = 998244353;
ll k;
int n, p, a[N], b[N];
inline int Pow(ll x, int y=P-2){
	ll ass=1;
	for(; y; y>>=1, x=x*x%P) if(y&1) ass=ass*x%P;
	return ass;
}
inline void NTT(int *f, int g){
	for(int i=0, j=0; i<p; ++i){
		if(i>j) swap(f[i], f[j]);
		for(int k=p>>1; (j^=k)<k; k>>=1);
	}
	for(int i=1; i<p; i<<=1){
		int w0=(g==1?Pow(3, (P-1)/i/2):Pow(Pow(3, (P-1)/i/2)));
		for(int j=0; j<p; j+=i<<1){
			int w=1;
			for(int k=j; k<j+i; ++k){
				int t=(ll)w*f[k+i]%P;
				f[k+i]=(P+f[k]-t)%P;
				f[k]=(f[k]+t)%P;
				w=(ll)w*w0%P;
			}
		}
	}
	if(g==-1){
		int I=Pow(p);
		rep(i, 0, p-1) f[i]=(ll)f[i]*I%P;
	}
}
int main() {
	read(n), read(k);
	rep(i, 0, n-1) read(a[i]);
	b[0]=1;
	rep(i, 1, n-1) b[i]=(k-1+i)%P*b[i-1]%P*Pow(i)%P;
	for(p=1; p<n*2-1; p<<=1);
	NTT(a, 1), NTT(b, 1);
	rep(i, 0, p-1) a[i]=(ll)a[i]*b[i]%P;
	NTT(a, -1);
	rep(i, 0, n-1) print(a[i]), print('\n');
	return flush(), 0;
}