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#include "poly/prefix_product_of_poly.hpp"
#pragma once
#include "linalg/matrix_mul.hpp"
#include "poly/lagrange_interpolate_iota.hpp"
// A[k-1]...A[0] を計算する
// アルゴリズム参考:https://github.com/noshi91/n91lib_rs/blob/master/src/algorithm/polynomial_matrix_prod.rs
// 実装参考:https://nyaannyaan.github.io/library/matrix/polynomial-matrix-prefix-prod.hpp
template <typename T>
vc<vc<T>> prefix_product_of_poly_matrix(vc<vc<vc<T>>>& A, ll k) {
int n = len(A);
using MAT = vc<vc<T>>;
auto shift = [&](vc<MAT>& G, T x) -> vc<MAT> {
int d = len(G);
vvv(T, H, d, n, n);
FOR(i, n) FOR(j, n) {
vc<T> g(d);
FOR(l, d) g[l] = G[l][i][j];
auto h = lagrange_interpolate_iota(g, x, d);
FOR(l, d) H[l][i][j] = h[l];
}
return H;
};
auto evaluate = [&](vc<T>& f, T x) -> T {
T res = 0;
T p = 1;
FOR(i, len(f)) {
res += f[i] * p;
p *= x;
}
return res;
};
ll deg = 1;
FOR(i, n) FOR(j, n) chmax(deg, len(A[i][j]) - 1);
vc<MAT> G(deg + 1);
ll v = 1;
while (deg * v * v < k) v *= 2;
T iv = T(1) / T(v);
FOR(i, len(G)) {
T x = T(v) * T(i);
vv(T, mat, n, n);
FOR(j, n) FOR(k, n) mat[j][k] = evaluate(A[j][k], x);
G[i] = mat;
}
for (ll w = 1; w != v; w *= 2) {
T W = w;
auto G1 = shift(G, W * iv);
auto G2 = shift(G, (W * T(deg) * T(v) + T(v)) * iv);
auto G3 = shift(G, (W * T(deg) * T(v) + T(v) + W) * iv);
FOR(i, w * deg + 1) {
G[i] = matrix_mul(G1[i], G[i]);
G2[i] = matrix_mul(G3[i], G2[i]);
}
copy(G2.begin(), G2.end() - 1, back_inserter(G));
}
vv(T, res, n, n);
FOR(i, n) res[i][i] = 1;
ll i = 0;
while (i + v <= k) res = matrix_mul(G[i / v], res), i += v;
while (i < k) {
vv(T, mat, n, n);
FOR(j, n) FOR(k, n) mat[j][k] = evaluate(A[j][k], i);
res = matrix_mul(mat, res);
++i;
}
return res;
}
// f[k-1]...f[0] を計算する
template <typename T>
T prefix_product_of_poly(vc<T>& f, ll k) {
vc<vc<vc<T>>> A(1);
A[0].resize(1);
A[0][0] = f;
auto res = prefix_product_of_poly_matrix(A, k);
return res[0][0];
}
#line 2 "mod/modint_common.hpp"
struct has_mod_impl {
template <class T>
static auto check(T &&x) -> decltype(x.get_mod(), std::true_type{});
template <class T>
static auto check(...) -> std::false_type;
};
template <class T>
class has_mod : public decltype(has_mod_impl::check<T>(std::declval<T>())) {};
template <typename mint>
mint inv(int n) {
static const int mod = mint::get_mod();
static vector<mint> dat = {0, 1};
assert(0 <= n);
if (n >= mod) n %= mod;
while (len(dat) <= n) {
int k = len(dat);
int q = (mod + k - 1) / k;
dat.eb(dat[k * q - mod] * mint::raw(q));
}
return dat[n];
}
template <typename mint>
mint fact(int n) {
static const int mod = mint::get_mod();
assert(0 <= n && n < mod);
static vector<mint> dat = {1, 1};
while (len(dat) <= n) dat.eb(dat[len(dat) - 1] * mint::raw(len(dat)));
return dat[n];
}
template <typename mint>
mint fact_inv(int n) {
static vector<mint> dat = {1, 1};
if (n < 0) return mint(0);
while (len(dat) <= n) dat.eb(dat[len(dat) - 1] * inv<mint>(len(dat)));
return dat[n];
}
template <class mint, class... Ts>
mint fact_invs(Ts... xs) {
return (mint(1) * ... * fact_inv<mint>(xs));
}
template <typename mint, class Head, class... Tail>
mint multinomial(Head &&head, Tail &&... tail) {
return fact<mint>(head) * fact_invs<mint>(std::forward<Tail>(tail)...);
}
template <typename mint>
mint C_dense(int n, int k) {
assert(n >= 0);
if (k < 0 || n < k) return 0;
static vvc<mint> C;
static int H = 0, W = 0;
auto calc = [&](int i, int j) -> mint {
if (i == 0) return (j == 0 ? mint(1) : mint(0));
return C[i - 1][j] + (j ? C[i - 1][j - 1] : 0);
};
if (W <= k) {
FOR(i, H) {
C[i].resize(k + 1);
FOR(j, W, k + 1) { C[i][j] = calc(i, j); }
}
W = k + 1;
}
if (H <= n) {
C.resize(n + 1);
FOR(i, H, n + 1) {
C[i].resize(W);
FOR(j, W) { C[i][j] = calc(i, j); }
}
H = n + 1;
}
return C[n][k];
}
template <typename mint, bool large = false, bool dense = false>
mint C(ll n, ll k) {
assert(n >= 0);
if (k < 0 || n < k) return 0;
if constexpr (dense) return C_dense<mint>(n, k);
if constexpr (!large) return multinomial<mint>(n, k, n - k);
k = min(k, n - k);
mint x(1);
FOR(i, k) x *= mint(n - i);
return x * fact_inv<mint>(k);
}
template <typename mint, bool large = false>
mint C_inv(ll n, ll k) {
assert(n >= 0);
assert(0 <= k && k <= n);
if (!large) return fact_inv<mint>(n) * fact<mint>(k) * fact<mint>(n - k);
return mint(1) / C<mint, 1>(n, k);
}
// [x^d](1-x)^{-n}
template <typename mint, bool large = false, bool dense = false>
mint C_negative(ll n, ll d) {
assert(n >= 0);
if (d < 0) return mint(0);
if (n == 0) { return (d == 0 ? mint(1) : mint(0)); }
return C<mint, large, dense>(n + d - 1, d);
}
#line 3 "mod/modint.hpp"
template <int mod>
struct modint {
static constexpr u32 umod = u32(mod);
static_assert(umod < u32(1) << 31);
u32 val;
static modint raw(u32 v) {
modint x;
x.val = v;
return x;
}
constexpr modint() : val(0) {}
constexpr modint(u32 x) : val(x % umod) {}
constexpr modint(u64 x) : val(x % umod) {}
constexpr modint(u128 x) : val(x % umod) {}
constexpr modint(int x) : val((x %= mod) < 0 ? x + mod : x){};
constexpr modint(ll x) : val((x %= mod) < 0 ? x + mod : x){};
constexpr modint(i128 x) : val((x %= mod) < 0 ? x + mod : x){};
bool operator<(const modint &other) const { return val < other.val; }
modint &operator+=(const modint &p) {
if ((val += p.val) >= umod) val -= umod;
return *this;
}
modint &operator-=(const modint &p) {
if ((val += umod - p.val) >= umod) val -= umod;
return *this;
}
modint &operator*=(const modint &p) {
val = u64(val) * p.val % umod;
return *this;
}
modint &operator/=(const modint &p) {
*this *= p.inverse();
return *this;
}
modint operator-() const { return modint::raw(val ? mod - val : u32(0)); }
modint operator+(const modint &p) const { return modint(*this) += p; }
modint operator-(const modint &p) const { return modint(*this) -= p; }
modint operator*(const modint &p) const { return modint(*this) *= p; }
modint operator/(const modint &p) const { return modint(*this) /= p; }
bool operator==(const modint &p) const { return val == p.val; }
bool operator!=(const modint &p) const { return val != p.val; }
modint inverse() const {
int a = val, b = mod, u = 1, v = 0, t;
while (b > 0) {
t = a / b;
swap(a -= t * b, b), swap(u -= t * v, v);
}
return modint(u);
}
modint pow(ll n) const {
if (n < 0) return inverse().pow(-n);
assert(n >= 0);
modint ret(1), mul(val);
while (n > 0) {
if (n & 1) ret *= mul;
mul *= mul;
n >>= 1;
}
return ret;
}
static constexpr int get_mod() { return mod; }
// (n, r), r は 1 の 2^n 乗根
static constexpr pair<int, int> ntt_info() {
if (mod == 120586241) return {20, 74066978};
if (mod == 167772161) return {25, 17};
if (mod == 469762049) return {26, 30};
if (mod == 754974721) return {24, 362};
if (mod == 880803841) return {23, 211};
if (mod == 943718401) return {22, 663003469};
if (mod == 998244353) return {23, 31};
if (mod == 1004535809) return {21, 582313106};
if (mod == 1012924417) return {21, 368093570};
return {-1, -1};
}
static constexpr bool can_ntt() { return ntt_info().fi != -1; }
};
#ifdef FASTIO
template <int mod>
void rd(modint<mod> &x) {
fastio::rd(x.val);
x.val %= mod;
// assert(0 <= x.val && x.val < mod);
}
template <int mod>
void wt(modint<mod> x) {
fastio::wt(x.val);
}
#endif
using modint107 = modint<1000000007>;
using modint998 = modint<998244353>;
#line 3 "linalg/matrix_mul.hpp"
template <class T, typename enable_if<has_mod<T>::value>::type* = nullptr>
vc<vc<T>> matrix_mul(const vc<vc<T>>& A, const vc<vc<T>>& B, int N1 = -1,
int N2 = -1, int N3 = -1) {
if (N1 == -1) { N1 = len(A), N2 = len(B), N3 = len(B[0]); }
vv(u32, b, N3, N2);
FOR(i, N2) FOR(j, N3) b[j][i] = B[i][j].val;
vv(T, C, N1, N3);
if ((T::get_mod() < (1 << 30)) && N2 <= 16) {
FOR(i, N1) FOR(j, N3) {
u64 sm = 0;
FOR(m, N2) sm += u64(A[i][m].val) * b[j][m];
C[i][j] = sm;
}
} else {
FOR(i, N1) FOR(j, N3) {
u128 sm = 0;
FOR(m, N2) sm += u64(A[i][m].val) * b[j][m];
C[i][j] = T::raw(sm % (T::get_mod()));
}
}
return C;
}
template <class T, typename enable_if<!has_mod<T>::value>::type* = nullptr>
vc<vc<T>> matrix_mul(const vc<vc<T>>& A, const vc<vc<T>>& B, int N1 = -1,
int N2 = -1, int N3 = -1) {
if (N1 == -1) { N1 = len(A), N2 = len(B), N3 = len(B[0]); }
vv(T, b, N2, N3);
FOR(i, N2) FOR(j, N3) b[j][i] = B[i][j];
vv(T, C, N1, N3);
FOR(n, N1) FOR(m, N2) FOR(k, N3) C[n][k] += A[n][m] * b[k][m];
return C;
}
// square-matrix defined as array
template <class T, int N,
typename enable_if<has_mod<T>::value>::type* = nullptr>
array<array<T, N>, N> matrix_mul(const array<array<T, N>, N>& A,
const array<array<T, N>, N>& B) {
array<array<T, N>, N> C{};
if ((T::get_mod() < (1 << 30)) && N <= 16) {
FOR(i, N) FOR(k, N) {
u64 sm = 0;
FOR(j, N) sm += u64(A[i][j].val) * (B[j][k].val);
C[i][k] = sm;
}
} else {
FOR(i, N) FOR(k, N) {
u128 sm = 0;
FOR(j, N) sm += u64(A[i][j].val) * (B[j][k].val);
C[i][k] = sm;
}
}
return C;
}
// square-matrix defined as array
template <class T, int N,
typename enable_if<!has_mod<T>::value>::type* = nullptr>
array<array<T, N>, N> matrix_mul(const array<array<T, N>, N>& A,
const array<array<T, N>, N>& B) {
array<array<T, N>, N> C{};
FOR(i, N) FOR(j, N) FOR(k, N) C[i][k] += A[i][j] * B[j][k];
return C;
}
#line 2 "alg/monoid/mul.hpp"
template <class T>
struct Monoid_Mul {
using value_type = T;
using X = T;
static constexpr X op(const X &x, const X &y) noexcept { return x * y; }
static constexpr X inverse(const X &x) noexcept { return X(1) / x; }
static constexpr X unit() { return X(1); }
static constexpr bool commute = true;
};
#line 1 "ds/sliding_window_aggregation.hpp"
template <class Monoid>
struct Sliding_Window_Aggregation {
using X = typename Monoid::value_type;
using value_type = X;
int sz = 0;
vc<X> dat;
vc<X> cum_l;
X cum_r;
Sliding_Window_Aggregation() : cum_l({Monoid::unit()}), cum_r(Monoid::unit()) {}
int size() { return sz; }
void push(X x) {
++sz;
cum_r = Monoid::op(cum_r, x);
dat.eb(x);
}
void pop() {
--sz;
cum_l.pop_back();
if (len(cum_l) == 0) {
cum_l = {Monoid::unit()};
cum_r = Monoid::unit();
while (len(dat) > 1) {
cum_l.eb(Monoid::op(dat.back(), cum_l.back()));
dat.pop_back();
}
dat.pop_back();
}
}
X lprod() { return cum_l.back(); }
X rprod() { return cum_r; }
X prod() { return Monoid::op(cum_l.back(), cum_r); }
};
// 定数倍は目に見えて遅くなるので、queue でよいときは使わない
template <class Monoid>
struct SWAG_deque {
using X = typename Monoid::value_type;
using value_type = X;
int sz;
vc<X> dat_l, dat_r;
vc<X> cum_l, cum_r;
SWAG_deque() : sz(0), cum_l({Monoid::unit()}), cum_r({Monoid::unit()}) {}
int size() { return sz; }
void push_back(X x) {
++sz;
dat_r.eb(x);
cum_r.eb(Monoid::op(cum_r.back(), x));
}
void push_front(X x) {
++sz;
dat_l.eb(x);
cum_l.eb(Monoid::op(x, cum_l.back()));
}
void push(X x) { push_back(x); }
void clear() {
sz = 0;
dat_l.clear(), dat_r.clear();
cum_l = {Monoid::unit()}, cum_r = {Monoid::unit()};
}
void pop_front() {
if (sz == 1) return clear();
if (dat_l.empty()) rebuild();
--sz;
dat_l.pop_back();
cum_l.pop_back();
}
void pop_back() {
if (sz == 1) return clear();
if (dat_r.empty()) rebuild();
--sz;
dat_r.pop_back();
cum_r.pop_back();
}
void pop() { pop_front(); }
X front() {
if (len(dat_l)) return dat_l.back();
return dat_r[0];
}
X lprod() { return cum_l.back(); }
X rprod() { return cum_r.back(); }
X prod() { return Monoid::op(cum_l.back(), cum_r.back()); }
X prod_all() { return prod(); }
private:
void rebuild() {
vc<X> X;
reverse(all(dat_l));
concat(X, dat_l, dat_r);
clear();
int m = len(X) / 2;
FOR_R(i, m) push_front(X[i]);
FOR(i, m, len(X)) push_back(X[i]);
assert(sz == len(X));
}
};
#line 2 "mod/mod_inv.hpp"
// long でも大丈夫
// (val * x - 1) が mod の倍数になるようにする
// 特に mod=0 なら x=0 が満たす
ll mod_inv(ll val, ll mod) {
if (mod == 0) return 0;
mod = abs(mod);
val %= mod;
if (val < 0) val += mod;
ll a = val, b = mod, u = 1, v = 0, t;
while (b > 0) {
t = a / b;
swap(a -= t * b, b), swap(u -= t * v, v);
}
if (u < 0) u += mod;
return u;
}
#line 2 "mod/crt3.hpp"
constexpr u32 mod_pow_constexpr(u64 a, u64 n, u32 mod) {
a %= mod;
u64 res = 1;
FOR(32) {
if (n & 1) res = res * a % mod;
a = a * a % mod, n /= 2;
}
return res;
}
template <typename T, u32 p0, u32 p1>
T CRT2(u64 a0, u64 a1) {
static_assert(p0 < p1);
static constexpr u64 x0_1 = mod_pow_constexpr(p0, p1 - 2, p1);
u64 c = (a1 - a0 + p1) * x0_1 % p1;
return a0 + c * p0;
}
template <typename T, u32 p0, u32 p1, u32 p2>
T CRT3(u64 a0, u64 a1, u64 a2) {
static_assert(p0 < p1 && p1 < p2);
static constexpr u64 x1 = mod_pow_constexpr(p0, p1 - 2, p1);
static constexpr u64 x2 = mod_pow_constexpr(u64(p0) * p1 % p2, p2 - 2, p2);
static constexpr u64 p01 = u64(p0) * p1;
u64 c = (a1 - a0 + p1) * x1 % p1;
u64 ans_1 = a0 + c * p0;
c = (a2 - ans_1 % p2 + p2) * x2 % p2;
return T(ans_1) + T(c) * T(p01);
}
template <typename T, u32 p0, u32 p1, u32 p2, u32 p3>
T CRT4(u64 a0, u64 a1, u64 a2, u64 a3) {
static_assert(p0 < p1 && p1 < p2 && p2 < p3);
static constexpr u64 x1 = mod_pow_constexpr(p0, p1 - 2, p1);
static constexpr u64 x2 = mod_pow_constexpr(u64(p0) * p1 % p2, p2 - 2, p2);
static constexpr u64 x3 = mod_pow_constexpr(u64(p0) * p1 % p3 * p2 % p3, p3 - 2, p3);
static constexpr u64 p01 = u64(p0) * p1;
u64 c = (a1 - a0 + p1) * x1 % p1;
u64 ans_1 = a0 + c * p0;
c = (a2 - ans_1 % p2 + p2) * x2 % p2;
u128 ans_2 = ans_1 + c * static_cast<u128>(p01);
c = (a3 - ans_2 % p3 + p3) * x3 % p3;
return T(ans_2) + T(c) * T(p01) * T(p2);
}
template <typename T, u32 p0, u32 p1, u32 p2, u32 p3, u32 p4>
T CRT5(u64 a0, u64 a1, u64 a2, u64 a3, u64 a4) {
static_assert(p0 < p1 && p1 < p2 && p2 < p3 && p3 < p4);
static constexpr u64 x1 = mod_pow_constexpr(p0, p1 - 2, p1);
static constexpr u64 x2 = mod_pow_constexpr(u64(p0) * p1 % p2, p2 - 2, p2);
static constexpr u64 x3 = mod_pow_constexpr(u64(p0) * p1 % p3 * p2 % p3, p3 - 2, p3);
static constexpr u64 x4 = mod_pow_constexpr(u64(p0) * p1 % p4 * p2 % p4 * p3 % p4, p4 - 2, p4);
static constexpr u64 p01 = u64(p0) * p1;
static constexpr u64 p23 = u64(p2) * p3;
u64 c = (a1 - a0 + p1) * x1 % p1;
u64 ans_1 = a0 + c * p0;
c = (a2 - ans_1 % p2 + p2) * x2 % p2;
u128 ans_2 = ans_1 + c * static_cast<u128>(p01);
c = static_cast<u64>(a3 - ans_2 % p3 + p3) * x3 % p3;
u128 ans_3 = ans_2 + static_cast<u128>(c * p2) * p01;
c = static_cast<u64>(a4 - ans_3 % p4 + p4) * x4 % p4;
return T(ans_3) + T(c) * T(p01) * T(p23);
}
#line 2 "poly/convolution_naive.hpp"
template <class T, typename enable_if<!has_mod<T>::value>::type* = nullptr>
vc<T> convolution_naive(const vc<T>& a, const vc<T>& b) {
int n = int(a.size()), m = int(b.size());
if (n > m) return convolution_naive<T>(b, a);
if (n == 0) return {};
vector<T> ans(n + m - 1);
FOR(i, n) FOR(j, m) ans[i + j] += a[i] * b[j];
return ans;
}
template <class T, typename enable_if<has_mod<T>::value>::type* = nullptr>
vc<T> convolution_naive(const vc<T>& a, const vc<T>& b) {
int n = int(a.size()), m = int(b.size());
if (n > m) return convolution_naive<T>(b, a);
if (n == 0) return {};
vc<T> ans(n + m - 1);
if (n <= 16 && (T::get_mod() < (1 << 30))) {
for (int k = 0; k < n + m - 1; ++k) {
int s = max(0, k - m + 1);
int t = min(n, k + 1);
u64 sm = 0;
for (int i = s; i < t; ++i) { sm += u64(a[i].val) * (b[k - i].val); }
ans[k] = sm;
}
} else {
for (int k = 0; k < n + m - 1; ++k) {
int s = max(0, k - m + 1);
int t = min(n, k + 1);
u128 sm = 0;
for (int i = s; i < t; ++i) { sm += u64(a[i].val) * (b[k - i].val); }
ans[k] = T::raw(sm % T::get_mod());
}
}
return ans;
}
#line 2 "poly/convolution_karatsuba.hpp"
// 任意の環でできる
template <typename T>
vc<T> convolution_karatsuba(const vc<T>& f, const vc<T>& g) {
const int thresh = 30;
if (min(len(f), len(g)) <= thresh) return convolution_naive(f, g);
int n = max(len(f), len(g));
int m = ceil(n, 2);
vc<T> f1, f2, g1, g2;
if (len(f) < m) f1 = f;
if (len(f) >= m) f1 = {f.begin(), f.begin() + m};
if (len(f) >= m) f2 = {f.begin() + m, f.end()};
if (len(g) < m) g1 = g;
if (len(g) >= m) g1 = {g.begin(), g.begin() + m};
if (len(g) >= m) g2 = {g.begin() + m, g.end()};
vc<T> a = convolution_karatsuba(f1, g1);
vc<T> b = convolution_karatsuba(f2, g2);
FOR(i, len(f2)) f1[i] += f2[i];
FOR(i, len(g2)) g1[i] += g2[i];
vc<T> c = convolution_karatsuba(f1, g1);
vc<T> F(len(f) + len(g) - 1);
assert(2 * m + len(b) <= len(F));
FOR(i, len(a)) F[i] += a[i], c[i] -= a[i];
FOR(i, len(b)) F[2 * m + i] += b[i], c[i] -= b[i];
if (c.back() == T(0)) c.pop_back();
FOR(i, len(c)) if (c[i] != T(0)) F[m + i] += c[i];
return F;
}
#line 2 "poly/ntt.hpp"
template <class mint>
void ntt(vector<mint>& a, bool inverse) {
assert(mint::can_ntt());
const int rank2 = mint::ntt_info().fi;
const int mod = mint::get_mod();
static array<mint, 30> root, iroot;
static array<mint, 30> rate2, irate2;
static array<mint, 30> rate3, irate3;
assert(rank2 != -1 && len(a) <= (1 << max(0, rank2)));
static bool prepared = 0;
if (!prepared) {
prepared = 1;
root[rank2] = mint::ntt_info().se;
iroot[rank2] = mint(1) / root[rank2];
FOR_R(i, rank2) {
root[i] = root[i + 1] * root[i + 1];
iroot[i] = iroot[i + 1] * iroot[i + 1];
}
mint prod = 1, iprod = 1;
for (int i = 0; i <= rank2 - 2; i++) {
rate2[i] = root[i + 2] * prod;
irate2[i] = iroot[i + 2] * iprod;
prod *= iroot[i + 2];
iprod *= root[i + 2];
}
prod = 1, iprod = 1;
for (int i = 0; i <= rank2 - 3; i++) {
rate3[i] = root[i + 3] * prod;
irate3[i] = iroot[i + 3] * iprod;
prod *= iroot[i + 3];
iprod *= root[i + 3];
}
}
int n = int(a.size());
int h = topbit(n);
assert(n == 1 << h);
if (!inverse) {
int len = 0;
while (len < h) {
if (h - len == 1) {
int p = 1 << (h - len - 1);
mint rot = 1;
FOR(s, 1 << len) {
int offset = s << (h - len);
FOR(i, p) {
auto l = a[i + offset];
auto r = a[i + offset + p] * rot;
a[i + offset] = l + r;
a[i + offset + p] = l - r;
}
rot *= rate2[topbit(~s & -~s)];
}
len++;
} else {
int p = 1 << (h - len - 2);
mint rot = 1, imag = root[2];
for (int s = 0; s < (1 << len); s++) {
mint rot2 = rot * rot;
mint rot3 = rot2 * rot;
int offset = s << (h - len);
for (int i = 0; i < p; i++) {
u64 mod2 = u64(mod) * mod;
u64 a0 = a[i + offset].val;
u64 a1 = u64(a[i + offset + p].val) * rot.val;
u64 a2 = u64(a[i + offset + 2 * p].val) * rot2.val;
u64 a3 = u64(a[i + offset + 3 * p].val) * rot3.val;
u64 a1na3imag = (a1 + mod2 - a3) % mod * imag.val;
u64 na2 = mod2 - a2;
a[i + offset] = a0 + a2 + a1 + a3;
a[i + offset + 1 * p] = a0 + a2 + (2 * mod2 - (a1 + a3));
a[i + offset + 2 * p] = a0 + na2 + a1na3imag;
a[i + offset + 3 * p] = a0 + na2 + (mod2 - a1na3imag);
}
rot *= rate3[topbit(~s & -~s)];
}
len += 2;
}
}
} else {
mint coef = mint(1) / mint(len(a));
FOR(i, len(a)) a[i] *= coef;
int len = h;
while (len) {
if (len == 1) {
int p = 1 << (h - len);
mint irot = 1;
FOR(s, 1 << (len - 1)) {
int offset = s << (h - len + 1);
FOR(i, p) {
u64 l = a[i + offset].val;
u64 r = a[i + offset + p].val;
a[i + offset] = l + r;
a[i + offset + p] = (mod + l - r) * irot.val;
}
irot *= irate2[topbit(~s & -~s)];
}
len--;
} else {
int p = 1 << (h - len);
mint irot = 1, iimag = iroot[2];
FOR(s, (1 << (len - 2))) {
mint irot2 = irot * irot;
mint irot3 = irot2 * irot;
int offset = s << (h - len + 2);
for (int i = 0; i < p; i++) {
u64 a0 = a[i + offset + 0 * p].val;
u64 a1 = a[i + offset + 1 * p].val;
u64 a2 = a[i + offset + 2 * p].val;
u64 a3 = a[i + offset + 3 * p].val;
u64 x = (mod + a2 - a3) * iimag.val % mod;
a[i + offset] = a0 + a1 + a2 + a3;
a[i + offset + 1 * p] = (a0 + mod - a1 + x) * irot.val;
a[i + offset + 2 * p] = (a0 + a1 + 2 * mod - a2 - a3) * irot2.val;
a[i + offset + 3 * p] = (a0 + 2 * mod - a1 - x) * irot3.val;
}
irot *= irate3[topbit(~s & -~s)];
}
len -= 2;
}
}
}
}
#line 8 "poly/convolution.hpp"
template <class mint>
vector<mint> convolution_ntt(vector<mint> a, vector<mint> b) {
if (a.empty() || b.empty()) return {};
int n = int(a.size()), m = int(b.size());
int sz = 1;
while (sz < n + m - 1) sz *= 2;
// sz = 2^k のときの高速化。分割統治的なやつで損しまくるので。
if ((n + m - 3) <= sz / 2) {
auto a_last = a.back(), b_last = b.back();
a.pop_back(), b.pop_back();
auto c = convolution(a, b);
c.resize(n + m - 1);
c[n + m - 2] = a_last * b_last;
FOR(i, len(a)) c[i + len(b)] += a[i] * b_last;
FOR(i, len(b)) c[i + len(a)] += b[i] * a_last;
return c;
}
a.resize(sz), b.resize(sz);
bool same = a == b;
ntt(a, 0);
if (same) {
b = a;
} else {
ntt(b, 0);
}
FOR(i, sz) a[i] *= b[i];
ntt(a, 1);
a.resize(n + m - 1);
return a;
}
template <typename mint>
vector<mint> convolution_garner(const vector<mint>& a, const vector<mint>& b) {
int n = len(a), m = len(b);
if (!n || !m) return {};
static constexpr int p0 = 167772161;
static constexpr int p1 = 469762049;
static constexpr int p2 = 754974721;
using mint0 = modint<p0>;
using mint1 = modint<p1>;
using mint2 = modint<p2>;
vc<mint0> a0(n), b0(m);
vc<mint1> a1(n), b1(m);
vc<mint2> a2(n), b2(m);
FOR(i, n) a0[i] = a[i].val, a1[i] = a[i].val, a2[i] = a[i].val;
FOR(i, m) b0[i] = b[i].val, b1[i] = b[i].val, b2[i] = b[i].val;
auto c0 = convolution_ntt<mint0>(a0, b0);
auto c1 = convolution_ntt<mint1>(a1, b1);
auto c2 = convolution_ntt<mint2>(a2, b2);
vc<mint> c(len(c0));
FOR(i, n + m - 1) { c[i] = CRT3<mint, p0, p1, p2>(c0[i].val, c1[i].val, c2[i].val); }
return c;
}
vector<ll> convolution(vector<ll> a, vector<ll> b) {
int n = len(a), m = len(b);
if (!n || !m) return {};
if (min(n, m) <= 2500) return convolution_naive(a, b);
ll mi_a = MIN(a), mi_b = MIN(b);
for (auto& x: a) x -= mi_a;
for (auto& x: b) x -= mi_b;
assert(MAX(a) * MAX(b) <= 1e18);
auto Ac = cumsum<ll>(a), Bc = cumsum<ll>(b);
vi res(n + m - 1);
for (int k = 0; k < n + m - 1; ++k) {
int s = max(0, k - m + 1);
int t = min(n, k + 1);
res[k] += (t - s) * mi_a * mi_b;
res[k] += mi_a * (Bc[k - s + 1] - Bc[k - t + 1]);
res[k] += mi_b * (Ac[t] - Ac[s]);
}
static constexpr u32 MOD1 = 1004535809;
static constexpr u32 MOD2 = 1012924417;
using mint1 = modint<MOD1>;
using mint2 = modint<MOD2>;
vc<mint1> a1(n), b1(m);
vc<mint2> a2(n), b2(m);
FOR(i, n) a1[i] = a[i], a2[i] = a[i];
FOR(i, m) b1[i] = b[i], b2[i] = b[i];
auto c1 = convolution_ntt<mint1>(a1, b1);
auto c2 = convolution_ntt<mint2>(a2, b2);
FOR(i, n + m - 1) { res[i] += CRT2<u64, MOD1, MOD2>(c1[i].val, c2[i].val); }
return res;
}
template <typename mint>
vc<mint> convolution(const vc<mint>& a, const vc<mint>& b) {
int n = len(a), m = len(b);
if (!n || !m) return {};
if (mint::can_ntt()) {
if (min(n, m) <= 50) return convolution_karatsuba<mint>(a, b);
return convolution_ntt(a, b);
}
if (min(n, m) <= 200) return convolution_karatsuba<mint>(a, b);
return convolution_garner(a, b);
}
#line 5 "poly/lagrange_interpolate_iota.hpp"
// Input: f(0), ..., f(n-1) and c. Return: f(c)
template <typename T, typename enable_if<has_mod<T>::value>::type * = nullptr>
T lagrange_interpolate_iota(vc<T> &f, T c) {
int n = len(f);
if (int(c.val) < n) return f[c.val];
auto a = f;
FOR(i, n) {
a[i] = a[i] * fact_inv<T>(i) * fact_inv<T>(n - 1 - i);
if ((n - 1 - i) & 1) a[i] = -a[i];
}
vc<T> lp(n + 1), rp(n + 1);
lp[0] = rp[n] = 1;
FOR(i, n) lp[i + 1] = lp[i] * (c - i);
FOR_R(i, n) rp[i] = rp[i + 1] * (c - i);
T ANS = 0;
FOR(i, n) ANS += a[i] * lp[i] * rp[i + 1];
return ANS;
}
// mod じゃない場合。かなり低次の多項式を想定している。O(n^2)
// Input: f(0), ..., f(n-1) and c. Return: f(c)
template <typename T, typename enable_if<!has_mod<T>::value>::type * = nullptr>
T lagrange_interpolate_iota(vc<T> &f, T c) {
const int LIM = 10;
int n = len(f);
assert(n < LIM);
// (-1)^{i-j} binom(i,j)
static vvc<int> C;
if (C.empty()) {
C.assign(LIM, vc<int>(LIM));
C[0][0] = 1;
FOR(n, 1, LIM) FOR(k, n + 1) {
C[n][k] += C[n - 1][k];
if (k) C[n][k] += C[n - 1][k - 1];
}
FOR(n, LIM) FOR(k, n + 1) if ((n + k) % 2) C[n][k] = -C[n][k];
}
// f(x) = sum a_i binom(x,i)
vc<T> a(n);
FOR(i, n) FOR(j, i + 1) { a[i] += f[j] * C[i][j]; }
T res = 0;
T b = 1;
FOR(i, n) {
res += a[i] * b;
b = b * (c - i) / (1 + i);
}
return res;
}
// Input: f(0), ..., f(n-1) and c, m
// Return: f(c), f(c+1), ..., f(c+m-1)
// Complexity: M(n, n + m)
template <typename mint>
vc<mint> lagrange_interpolate_iota(vc<mint> &f, mint c, int m) {
if (m <= 60) {
vc<mint> ANS(m);
FOR(i, m) ANS[i] = lagrange_interpolate_iota(f, c + mint(i));
return ANS;
}
ll n = len(f);
auto a = f;
FOR(i, n) {
a[i] = a[i] * fact_inv<mint>(i) * fact_inv<mint>(n - 1 - i);
if ((n - 1 - i) & 1) a[i] = -a[i];
}
// x = c, c+1, ... に対して a0/x + a1/(x-1) + ... を求めておく
vc<mint> b(n + m - 1);
FOR(i, n + m - 1) b[i] = mint(1) / (c + mint(i - n + 1));
a = convolution(a, b);
Sliding_Window_Aggregation<Monoid_Mul<mint>> swag;
vc<mint> ANS(m);
ll L = 0, R = 0;
FOR(i, m) {
while (L < i) { swag.pop(), ++L; }
while (R - L < n) { swag.push(c + mint((R++) - n + 1)); }
auto coef = swag.prod();
if (coef == 0) {
ANS[i] = f[(c + i).val];
} else {
ANS[i] = a[i + n - 1] * coef;
}
}
return ANS;
}
#line 4 "poly/prefix_product_of_poly.hpp"
// A[k-1]...A[0] を計算する
// アルゴリズム参考:https://github.com/noshi91/n91lib_rs/blob/master/src/algorithm/polynomial_matrix_prod.rs
// 実装参考:https://nyaannyaan.github.io/library/matrix/polynomial-matrix-prefix-prod.hpp
template <typename T>
vc<vc<T>> prefix_product_of_poly_matrix(vc<vc<vc<T>>>& A, ll k) {
int n = len(A);
using MAT = vc<vc<T>>;
auto shift = [&](vc<MAT>& G, T x) -> vc<MAT> {
int d = len(G);
vvv(T, H, d, n, n);
FOR(i, n) FOR(j, n) {
vc<T> g(d);
FOR(l, d) g[l] = G[l][i][j];
auto h = lagrange_interpolate_iota(g, x, d);
FOR(l, d) H[l][i][j] = h[l];
}
return H;
};
auto evaluate = [&](vc<T>& f, T x) -> T {
T res = 0;
T p = 1;
FOR(i, len(f)) {
res += f[i] * p;
p *= x;
}
return res;
};
ll deg = 1;
FOR(i, n) FOR(j, n) chmax(deg, len(A[i][j]) - 1);
vc<MAT> G(deg + 1);
ll v = 1;
while (deg * v * v < k) v *= 2;
T iv = T(1) / T(v);
FOR(i, len(G)) {
T x = T(v) * T(i);
vv(T, mat, n, n);
FOR(j, n) FOR(k, n) mat[j][k] = evaluate(A[j][k], x);
G[i] = mat;
}
for (ll w = 1; w != v; w *= 2) {
T W = w;
auto G1 = shift(G, W * iv);
auto G2 = shift(G, (W * T(deg) * T(v) + T(v)) * iv);
auto G3 = shift(G, (W * T(deg) * T(v) + T(v) + W) * iv);
FOR(i, w * deg + 1) {
G[i] = matrix_mul(G1[i], G[i]);
G2[i] = matrix_mul(G3[i], G2[i]);
}
copy(G2.begin(), G2.end() - 1, back_inserter(G));
}
vv(T, res, n, n);
FOR(i, n) res[i][i] = 1;
ll i = 0;
while (i + v <= k) res = matrix_mul(G[i / v], res), i += v;
while (i < k) {
vv(T, mat, n, n);
FOR(j, n) FOR(k, n) mat[j][k] = evaluate(A[j][k], i);
res = matrix_mul(mat, res);
++i;
}
return res;
}
// f[k-1]...f[0] を計算する
template <typename T>
T prefix_product_of_poly(vc<T>& f, ll k) {
vc<vc<vc<T>>> A(1);
A[0].resize(1);
A[0][0] = f;
auto res = prefix_product_of_poly_matrix(A, k);
return res[0][0];
}