[模板] 常系数齐次线性递推

多项式取模优化。

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#include <iostream>
#include <cstring>
#include <cstdio>
#include <algorithm>

using namespace std;

const int maxn = 32010;
const int mod = 998244353;
const int g = 3;

int n, k;
int f[maxn], a[maxn];

int qpow(int x, int y) {
int ret = 1;
while (y) {
if (y & 1) ret = 1LL*ret*x%mod;
x = 1LL*x*x%mod;
y >>= 1;
}
return ret;
}

struct poly {
int *a, len;
poly(int l_ = 0) {
len = l_;
a = new int[len];
for (int i = 0; i < len; i++)
a[i] = 0;
}
} P; // 特征多项式

int wa[maxn*8], wb[maxn*8], wc[maxn*8], rev[maxn*8];

void ntt(int *a, int l, int ty) {
for (int i = 1; i < (1<<l); i++) rev[i] = ((rev[i>>1]>>1) | ((i&1)<<(l-1)));
for (int i = 0; i < (1<<l); i++) if (i < rev[i]) swap(a[i], a[rev[i]]);
for (int len = 2; len <= (1<<l); len <<= 1) {
int wn = qpow(g, (mod-1)/len);
for (int s = 0; s < (1<<l); s += len) {
int w = 1;
for (int i = s; i < s + (len>>1); ++ i) {
int v1 = a[i], v2 = 1LL*w*a[i+(len>>1)]%mod;
a[i] = (v1 + v2) % mod;
a[i + (len >> 1)] = (v1 + mod - v2) % mod;
w = 1LL*w*wn%mod;
}
}
}
if (ty == -1) {
int inv = qpow((1<<l), mod-2);
for (int i = 0; i < (1<<l); i++) a[i] = 1LL*a[i]*inv%mod;
for (int i = 1; i < (1<<(l-1)); i++) swap(a[i], a[(1<<l)-i]);
}
}

poly operator*(const poly &p1, const poly &p2) {
poly ret(p1.len + p2.len - 1);
int l = 0; while ((1<<l) < ret.len) ++ l;
for (int i = 0; i < (1<<l); i++) wa[i] = wb[i] = 0;
for (int i = 0; i < p1.len; i++) wa[i] = p1.a[i];
for (int i = 0; i < p2.len; i++) wb[i] = p2.a[i];
ntt(wa, l, 1); ntt(wb, l, 1);
for (int i = 0; i < (1<<l); i++) wc[i] = 1LL*wa[i]*wb[i]%mod;
ntt(wc, l, -1);
for (int i = 0; i < ret.len; i++) ret.a[i] = wc[i];
return ret;
}

poly polyInv(const poly &p) {
if (p.len == 1) {
poly ret(1);
ret.a[0] = qpow(p.a[0], mod-2);
return ret;
}
int nlen = (p.len + 1) / 2;
poly np(nlen); for (int i = 0; i < nlen; i++) np.a[i] = p.a[i];
poly f0 = polyInv(np);
poly t1 = p*f0;
poly t2(p.len);
for (int i = 0; i < p.len; i++) {
if (i < t1.len) {
t2.a[i] = (mod - t1.a[i]) % mod;
}
}
t2.a[0] = (t2.a[0] + 2) % mod;
poly res = f0*t2;
poly ret(p.len);
for (int i = 0; i < p.len; i++) ret.a[i] = res.a[i];
return ret;
}

poly polyMod(const poly &p, const poly &q) {
if (p.len < q.len) return p;
poly rp(p.len-q.len+1), rq(p.len-q.len+1);
for (int i = 0; i < p.len; i++) if (q.len-1-i < rp.len) rp.a[p.len-1-i] = p.a[i];
for (int i = 0; i < q.len; i++) if (q.len-1-i < rq.len) rq.a[q.len-1-i] = q.a[i];
poly t1 = rp*polyInv(rq);
poly t2(p.len-q.len+1);
for (int i = 0; i < t2.len; i++) {
t2.a[i] = t1.a[p.len-q.len-i];
}
poly t3 = t2*q;
poly ret(q.len-1);
for (int i = 0; i < ret.len; i++) ret.a[i] = (p.a[i] + mod - t3.a[i]) % mod;
return ret;
}

poly calMod(int y) {
poly ret(1); ret.a[0] = 1;
poly x(2); x.a[1] = 1;
while (y) {
if (y & 1) ret = polyMod(ret * x, P);
x = polyMod(x * x, P);
y >>= 1;
}
return ret;
}

void polyPrint(const poly &p) {
printf("%d :\n", p.len);
for (int i = 0; i < p.len; i++) {
printf("%d ", p.a[i]);
}
printf("\n");
}

int main() {
scanf("%d%d", &n, &k);
for (int i = 1; i <= k; i++) {scanf("%d", &a[i]); a[i] = (a[i] + mod) % mod;}
for (int i = 0; i < k; i++) {scanf("%d", &f[i]); f[i] = (f[i] + mod) % mod;}
//乘以 A^n
int ans = 0;
P = poly(k+1);
for (int i = 1; i <= k; i++) P.a[k-i] = (mod-a[i])%mod;
P.a[k] = 1;
//polyPrint(P);
poly pol = calMod(n);
//polyPrint(pol);
for (int i = 0; i < pol.len; i++) {
ans = (ans + 1LL*pol.a[i]*f[i]%mod)%mod;
}
printf("%d\n", ans);
return 0;
}