# LOJ 556: 「Antileaf's Round」咱们去烧菜吧

### 题意简述：

$m$ 种物品，第 $i$ 种物品的体积为 $a_i$，数量为 $b_i$

$1\le n,m\le 10^5$

### 题解：

\begin{aligned}\mathbf{OGF}\{\mathrm{Ans}\}&=\prod_{i=1}^{m}\sum_{j=0}^{b_i}x^{j\cdot a_i}\\&=\prod_{i=1}^{m}\frac{1-x^{a_i(b_i+1)}}{1-x^{a_i}}\end{aligned}

\begin{aligned}\ln(\mathbf{OGF}\{\mathrm{Ans}\})&=\sum_{i=1}^{m}\left(\ln\left(\frac{1}{1-x^{a_i}}\right)-\ln\left(\frac{1}{1-x^{a_i(b_i+1)}}\right)\right)\end{aligned}

\begin{aligned}\mathbf{OGF}\{\mathrm{Ans}\}&=\exp\left(\sum_{i=1}^{m}\left(\sum_{j=1}^{\infty}\frac{x^{ja_i}}{j}-\sum_{j=1}^{\infty}\frac{x^{ja_i(b_i+1)}}{j}\right)\right)\end{aligned}

#include <cstdio>
#include <algorithm>

typedef long long LL;
const int Mod = 998244353;
const int G = 3, iG = 332748118;
const int MS = 1 << 18;
const int MN = 100005;

inline int qPow(int b, int e) {
int a = 1;
for (; e; e >>= 1, b = (LL)b * b % Mod)
if (e & 1) a = (LL)a * b % Mod;
return a;
}

inline int gInv(int b) { return qPow(b, Mod - 2); }

int Inv[MS], Fac[MS], iFac[MS];

inline void Init(int N) {
Fac[0] = 1;
for (int i = 1; i < N; ++i) Fac[i] = (LL)Fac[i - 1] * i % Mod;
iFac[N - 1] = gInv(Fac[N - 1]);
for (int i = N - 1; i >= 1; --i) iFac[i - 1] = (LL)iFac[i] * i % Mod;
for (int i = 1; i < N; ++i) Inv[i] = (LL)Fac[i - 1] * iFac[i] % Mod;
}

int Sz, InvSz, R[MS];

inline int getB(int N) { int Bt = 0; while (1 << Bt < N) ++Bt; return Bt; }

inline void InitFNTT(int N) {
int Bt = getB(N);
if (Sz == (1 << Bt)) return ;
Sz = 1 << Bt, InvSz = Mod - (Mod - 1) / Sz;
for (int i = 1; i < Sz; ++i) R[i] = R[i >> 1] >> 1 | (i & 1) << (Bt - 1);
}

inline void FNTT(int *A, int Ty) {
for (int i = 0; i < Sz; ++i) if (R[i] < i) std::swap(A[R[i]], A[i]);
for (int j = 1, j2 = 2; j < Sz; j <<= 1, j2 <<= 1) {
int wn = qPow(~Ty ? G : iG, (Mod - 1) / j2), w, X, Y;
for (int i = 0, k; i < Sz; i += j2) {
for (k = 0, w = 1; k < j; ++k, w = (LL)w * wn % Mod) {
X = A[i + k], Y = (LL)w * A[i + j + k] % Mod;
A[i + k] -= (A[i + k] = X + Y) >= Mod ? Mod : 0;
A[i + j + k] += (A[i + j + k] = X - Y) < 0 ? Mod : 0;
}
}
}
if (!~Ty) for (int i = 0; i < Sz; ++i) A[i] = (LL)A[i] * InvSz % Mod;
}

inline void PolyInv(int *_A, int N, int *_B) {
static int A[MS], B[MS], tA[MS], tB[MS];
for (int i = 0; i < N; ++i) A[i] = _A[i];
for (int i = N, B = getB(N); i < 1 << B; ++i) A[i] = 0;
B[0] = gInv(A[0]);
for (int L = 1; L < N; L <<= 1) {
int L2 = L << 1, L4 = L << 2;
InitFNTT(L4);
for (int i = 0; i < L2; ++i) tA[i] = A[i];
for (int i = L2; i < Sz; ++i) tA[i] = 0;
for (int i = 0; i < L; ++i) tB[i] = B[i];
for (int i = L; i < Sz; ++i) tB[i] = 0;
FNTT(tA, 1), FNTT(tB, 1);
for (int i = 0; i < Sz; ++i) tB[i] = tB[i] * (2 - (LL)tA[i] * tB[i] % Mod + Mod) % Mod;
FNTT(tB, -1);
for (int i = 0; i < L2; ++i) B[i] = tB[i];
}
for (int i = 0; i < N; ++i) _B[i] = B[i];
}

inline void PolyLn(int *_A, int N, int *_B) {
static int tA[MS], tB[MS];
for (int i = 1; i < N; ++i) tA[i - 1] = (LL)_A[i] * i % Mod;
PolyInv(_A, N - 1, tB);
InitFNTT(N + N - 3);
for (int i = N - 1; i < Sz; ++i) tA[i] = 0;
for (int i = N - 1; i < Sz; ++i) tB[i] = 0;
FNTT(tA, 1), FNTT(tB, 1);
for (int i = 0; i < Sz; ++i) tA[i] = (LL)tA[i] * tB[i] % Mod;
FNTT(tA, -1);
_B[0] = 0;
for (int i = 1; i < N; ++i) _B[i] = (LL)tA[i - 1] * Inv[i] % Mod;
}

inline void PolyExp(int *_A, int N, int *_B) {
static int A[MS], B[MS], tA[MS], tB[MS];
for (int i = 0; i < N; ++i) A[i] = _A[i];
for (int i = N, B = getB(N); i < 1 << B; ++i) A[i] = 0;
B[0] = 1;
for (int L = 1; L < N; L <<= 1) {
int L2 = L << 1, L4 = L << 2;
for (int i = L; i < L2; ++i) B[i] = 0;
PolyLn(B, L2, tA);
InitFNTT(L4);
for (int i = 0; i < L2; ++i) tA[i] = (!i + A[i] - tA[i] + Mod) % Mod;
for (int i = L2; i < Sz; ++i) tA[i] = 0;
for (int i = 0; i < L; ++i) tB[i] = B[i];
for (int i = L; i < Sz; ++i) tB[i] = 0;
FNTT(tA, 1), FNTT(tB, 1);
for (int i = 0; i < Sz; ++i) tA[i] = (LL)tA[i] * tB[i] % Mod;
FNTT(tA, -1);
for (int i = 0; i < L2; ++i) B[i] = tA[i];
}
for (int i = 0; i < N; ++i) _B[i] = B[i];
}

int N, M, buk[MN];
int A[MS], B[MS];

int main() {
Init(MS);
scanf("%d%d", &N, &M);
for (int i = 1; i <= M; ++i) {
int A, B;
scanf("%d%d", &A, &B);
if (!A) continue;
if (!B) B = N;
if (A <= N) ++buk[A];
if (B + 1 <= N / A) --buk[A * (B + 1)];
}
Init(MS);
for (int i = 1; i <= N; ++i) if (buk[i]) {
if (buk[i] < 0) buk[i] += Mod;
for (int j = i, k = 1; j <= N; j += i, ++k)
A[j] = (A[j] + (LL)buk[i] * Inv[k]) % Mod;
}
PolyExp(A, N + 1, B);
for (int i = 1; i <= N; ++i) printf("%d\n", B[i]);
return 0;
}

posted @ 2019-04-27 20:36  粉兔  阅读(...)  评论(...编辑  收藏