結果
| 問題 | No.2485 Add to Variables (Another) |
| ユーザー |
 yuto1115
|
| 提出日時 | 2023-09-22 22:15:00 |
| 言語 | C++17(gcc12) (gcc 12.3.0 + boost 1.87.0) |
| 結果 |
AC
|
| 実行時間 | 59 ms / 2,000 ms |
| コード長 | 26,633 bytes |
| コンパイル時間 | 2,327 ms |
| コンパイル使用メモリ | 222,488 KB |
| 実行使用メモリ | 34,652 KB |
| 最終ジャッジ日時 | 2024-07-08 12:58:42 |
| 合計ジャッジ時間 | 4,883 ms |
|
ジャッジサーバーID （参考情報） |
judge5 / judge1 |
（要ログイン）
| ファイルパターン | 結果 |
|---|---|
| sample | AC * 5 |
| other | AC * 39 |
ソースコード
#include<bits/stdc++.h>
#define overload4(_1, _2, _3, _4, name, ...) name
#define rep1(i, n) for (ll i = 0; i < ll(n); ++i)
#define rep2(i, s, n) for (ll i = ll(s); i < ll(n); ++i)
#define rep3(i, s, n, d) for(ll i = ll(s); i < ll(n); i+=d)
#define rep(...) overload4(__VA_ARGS__,rep3,rep2,rep1)(__VA_ARGS__)
#define rrep1(i, n) for (ll i = ll(n)-1; i >= 0; i--)
#define rrep2(i, n, t) for (ll i = ll(n)-1; i >= (ll)t; i--)
#define rrep3(i, n, t, d) for (ll i = ll(n)-1; i >= (ll)t; i-=d)
#define rrep(...) overload4(__VA_ARGS__,rrep3,rrep2,rrep1)(__VA_ARGS__)
#define all(a) a.begin(),a.end()
#define rall(a) a.rbegin(),a.rend()
#define SUM(a) accumulate(all(a),0LL)
#define MIN(a) *min_element(all(a))
#define MAX(a) *max_element(all(a))
#define SORT(a) sort(all(a));
#define REV(a) reverse(all(a));
#define SZ(a) int(a.size())
#define popcount(x) __builtin_popcountll(x)
#define pf push_front
#define pb push_back
#define ef emplace_front
#define eb emplace_back
#define ppf pop_front
#define ppb pop_back
#ifdef __LOCAL
#define debug(...) { cout << #__VA_ARGS__; cout << ": "; print(__VA_ARGS__); cout << flush; }
#else
#define debug(...) void(0);
#endif
#define INT(...) int __VA_ARGS__;scan(__VA_ARGS__)
#define LL(...) ll __VA_ARGS__;scan(__VA_ARGS__)
#define STR(...) string __VA_ARGS__;scan(__VA_ARGS__)
#define CHR(...) char __VA_ARGS__;scan(__VA_ARGS__)
#define DBL(...) double __VA_ARGS__;scan(__VA_ARGS__)
#define LD(...) ld __VA_ARGS__;scan(__VA_ARGS__)
using namespace std;
using ll = long long;
using ld = long double;
using P = pair<int, int>;
using LP = pair<ll, ll>;
using vi = vector<int>;
using vvi = vector<vi>;
using vvvi = vector<vvi>;
using vl = vector<ll>;
using vvl = vector<vl>;
using vvvl = vector<vvl>;
using vd = vector<double>;
using vvd = vector<vd>;
using vs = vector<string>;
using vc = vector<char>;
using vvc = vector<vc>;
using vb = vector<bool>;
using vvb = vector<vb>;
using vp = vector<P>;
using vvp = vector<vp>;
using vlp = vector<LP>;
using vvlp = vector<vlp>;
template<class T>
using PQ = priority_queue<pair<T, int>, vector<pair<T, int>>, greater<pair<T, int>>>;
template<class S, class T>
istream &operator>>(istream &is, pair<S, T> &p) { return is >> p.first >> p.second; }
template<class S, class T>
ostream &operator<<(ostream &os, const pair<S, T> &p) { return os << '{' << p.first << ", " << p.second << '}'; }
template<class S, class T, class U>
istream &operator>>(istream &is, tuple<S, T, U> &t) { return is >> get<0>(t) >> get<1>(t) >> get<2>(t); }
template<class S, class T, class U>
ostream &operator<<(ostream &os, const tuple<S, T, U> &t) {
return os << '{' << get<0>(t) << ", " << get<1>(t) << ", " << get<2>(t) << '}';
}
template<class T>
istream &operator>>(istream &is, vector<T> &v) {
for (T &t: v) { is >> t; }
return is;
}
template<class T>
ostream &operator<<(ostream &os, const vector<T> &v) {
os << '[';
rep(i, v.size()) os << v[i] << (i == int(v.size() - 1) ? "" : ", ");
return os << ']';
}
template<class T>
ostream &operator<<(ostream &os, const deque<T> &v) {
os << '[';
rep(i, v.size()) os << v[i] << (i == int(v.size() - 1) ? "" : ", ");
return os << ']';
}
template<class T>
ostream &operator<<(ostream &os, const set<T> &st) {
os << '{';
auto it = st.begin();
while (it != st.end()) {
os << (it == st.begin() ? "" : ", ") << *it;
it++;
}
return os << '}';
}
template<class T>
ostream &operator<<(ostream &os, const multiset<T> &st) {
os << '{';
auto it = st.begin();
while (it != st.end()) {
os << (it == st.begin() ? "" : ", ") << *it;
it++;
}
return os << '}';
}
template<class T, class U>
ostream &operator<<(ostream &os, const map<T, U> &mp) {
os << '{';
auto it = mp.begin();
while (it != mp.end()) {
os << (it == mp.begin() ? "" : ", ") << *it;
it++;
}
return os << '}';
}
template<class T>
void vecout(const vector<T> &v, char div = '\n') {
rep(i, v.size()) cout << v[i] << (i == int(v.size() - 1) ? '\n' : div);
}
template<class T>
bool constexpr chmin(T &a, T b) {
if (a > b) {
a = b;
return true;
}
return false;
}
template<class T>
bool constexpr chmax(T &a, T b) {
if (a < b) {
a = b;
return true;
}
return false;
}
void scan() {}
template<class Head, class... Tail>
void scan(Head &head, Tail &... tail) {
cin >> head;
scan(tail...);
}
template<class T>
void print(const T &t) { cout << t << '\n'; }
template<class Head, class... Tail>
void print(const Head &head, const Tail &... tail) {
cout << head << ' ';
print(tail...);
}
template<class... T>
void fin(const T &... a) {
print(a...);
exit(0);
}
template<class T>
vector<T> &operator+=(vector<T> &v, T x) {
for (T &t: v) t += x;
return v;
}
template<class T>
vector<T> &operator-=(vector<T> &v, T x) {
for (T &t: v) t -= x;
return v;
}
template<class T>
vector<T> &operator*=(vector<T> &v, T x) {
for (T &t: v) t *= x;
return v;
}
template<class T>
vector<T> &operator/=(vector<T> &v, T x) {
for (T &t: v) t /= x;
return v;
}
struct Init_io {
Init_io() {
ios::sync_with_stdio(false);
cin.tie(nullptr);
cout.tie(nullptr);
cout << boolalpha << fixed << setprecision(15);
cerr << boolalpha << fixed << setprecision(15);
}
} init_io;
const string yes[] = {"no", "yes"};
const string Yes[] = {"No", "Yes"};
const string YES[] = {"NO", "YES"};
const int inf = 1001001001;
const ll linf = 1001001001001001001;
void rearrange(const vi &) {}
template<class T, class... Tail>
void rearrange(const vi &ord, vector<T> &head, Tail &...tail) {
assert(ord.size() == head.size());
vector<T> ori = head;
rep(i, ord.size()) head[i] = ori[ord[i]];
rearrange(ord, tail...);
}
template<class T, class... Tail>
void sort_by(vector<T> &head, Tail &... tail) {
vi ord(head.size());
iota(all(ord), 0);
sort(all(ord), [&](int i, int j) { return head[i] < head[j]; });
rearrange(ord, head, tail...);
}
bool in_rect(int i, int j, int h, int w) {
return 0 <= i and i < h and 0 <= j and j < w;
}
template<class T, class S>
vector<T> cumsum(const vector<S> &v, bool shift_one = true) {
int n = v.size();
vector<T> res;
if (shift_one) {
res.resize(n + 1);
rep(i, n) res[i + 1] = res[i] + v[i];
} else {
res.resize(n);
if (n) {
res[0] = v[0];
rep(i, 1, n) res[i] = res[i - 1] + v[i];
}
}
return res;
}
vvi graph(int n, int m, bool directed = false, int origin = 1) {
vvi G(n);
rep(_, m) {
INT(u, v);
u -= origin, v -= origin;
G[u].pb(v);
if (!directed) G[v].pb(u);
}
return G;
}
template<class T>
vector<vector<pair<int, T>>> weighted_graph(int n, int m, bool directed = false, int origin = 1) {
vector<vector<pair<int, T>>> G(n);
rep(_, m) {
int u, v;
T w;
scan(u, v, w);
u -= origin, v -= origin;
G[u].eb(v, w);
if (!directed) G[v].eb(u, w);
}
return G;
}
template<int mod>
class modint {
ll x;
public:
constexpr modint(ll x = 0) : x((x % mod + mod) % mod) {}
static constexpr int get_mod() { return mod; }
constexpr int val() const { return x; }
constexpr modint operator-() const { return modint(-x); }
constexpr modint &operator+=(const modint &a) {
if ((x += a.val()) >= mod) x -= mod;
return *this;
}
constexpr modint &operator++() { return *this += 1; }
constexpr modint &operator-=(const modint &a) {
if ((x += mod - a.val()) >= mod) x -= mod;
return *this;
}
constexpr modint &operator--() { return *this -= 1; }
constexpr modint
&
operator*=(const modint &a) {
(x *= a.val()) %= mod;
return *this;
}
constexpr modint
operator+(const modint &a) const {
modint res(*this);
return res += a;
}
constexpr modint
operator-(const modint &a) const {
modint res(*this);
return res -= a;
}
constexpr modint
operator*(const modint &a) const {
modint res(*this);
return res *= a;
}
constexpr modint
pow(ll
t) const {
modint res = 1, a(*this);
while (t > 0) {
if (t & 1) res *= a;
t >>= 1;
a *= a;
}
return res;
}
template<int m>
friend istream &operator>>(istream &, modint<m> &);
// for prime mod
constexpr modint
inv() const { return pow(mod - 2); }
constexpr modint
&
operator/=(const modint &a) { return *this *= a.inv(); }
constexpr modint operator/(const modint &a) const {
modint res(*this);
return res /= a;
}
// constraints : mod = 2 or val = 0 or val^((mod-1)/2) ≡ 1
// mod is prime
// time complexity : O(log^2 p)
// reference : https://nyaannyaan.github.io/library/modulo/mod-sqrt.hpp
modint sqrt() const {
if (x < 2) return x;
assert(this->pow((mod - 1) >> 1).val() == 1);
modint b = 1;
while (b.pow((mod - 1) >> 1).val() == 1) b += 1;
ll m = mod - 1, e = 0;
while (~m & 1) m >>= 1, e += 1;
modint X = this->pow((m - 1) >> 1);
modint Y = (*this) * X * X;
X *= *this;
modint Z = b.pow(m);
while (Y.val() != 1) {
ll j = 0;
modint t = Y;
while (t.val() != 1) {
j += 1;
t *= t;
}
Z = Z.pow(1LL << (e - j - 1));
X *= Z;
Z *= Z;
Y *= Z;
e = j;
}
return X;
}
};
using modint998244353 = modint<998244353>;
using modint1000000007 = modint<1000000007>;
template<int mod>
istream &operator>>(istream &is, modint<mod> &a) { return is >> a.x; }
template<int mod>
constexpr ostream &operator<<(ostream &os, const modint<mod> &a) { return os << a.val(); }
template<int mod>
constexpr bool operator==(const modint<mod> &a, const modint<mod> &b) { return a.val() == b.val(); }
template<int mod>
constexpr bool operator!=(const modint<mod> &a, const modint<mod> &b) { return a.val() != b.val(); }
template<int mod>
constexpr modint<mod> &operator++(modint<mod> &a) {
return a += 1;
}
template<int mod>
constexpr modint<mod> &operator--(modint<mod> &a) {
return a -= 1;
}
using mint = modint998244353;
using vm = vector<mint>;
using vvm = vector<vm>;
class NTT {
int pr;
constexpr ll pow_mod(ll x, ll n, int m) {
if (m == 1) return 0;
ll res = 1;
ll now = x % m;
while (n > 0) {
if (n & 1) res = (res * now) % m;
now = (now * now) % m;
n >>= 1;
}
return res;
}
constexpr int primitive_root(int mod) {
if (mod == 2) return 1;
if (mod == 167772161) return 3;
if (mod == 469762049) return 3;
if (mod == 754974721) return 11;
if (mod == 998244353) return 3;
int divs[20] = {};
divs[0] = 2;
int cnt = 1;
int x = (mod - 1) / 2;
while (x % 2 == 0) x /= 2;
for (int i = 3; (ll) i * i <= x; i += 2) {
if (x % i == 0) {
divs[cnt++] = i;
while (x % i == 0) {
x /= i;
}
}
}
if (x > 1) divs[cnt++] = x;
for (int g = 2;; g++) {
bool ok = true;
for (int i = 0; i < cnt; i++) {
if (pow_mod(g, (mod - 1) / divs[i], mod) == 1) {
ok = false;
break;
}
}
if (ok) return g;
}
}
public:
NTT() { init(mint::get_mod()); }
mint sum_e[30]; // sum_e[i] = ies[0] * ... * ies[i - 1] * es[i]
mint sum_ie[30]; // sum_ie[i] = es[0] * ... * es[i - 1] * ies[i]
void init(int mod) {
pr = primitive_root(mod);
mint es[30], ies[30]; // es[i]^(2^(2+i)) == 1
int cnt2 = __builtin_ctz(mint::get_mod() - 1);
mint e = mint(pr).pow((mint::get_mod() - 1) >> cnt2), ie = e.inv();
for (int i = cnt2; i >= 2; i--) {
// e^(2^i) == 1
es[i - 2] = e;
ies[i - 2] = ie;
e *= e;
ie *= ie;
}
mint now = 1;
for (int i = 0; i <= cnt2 - 2; i++) {
sum_e[i] = es[i] * now;
now *= ies[i];
}
now = 1;
for (int i = 0; i <= cnt2 - 2; i++) {
sum_ie[i] = ies[i] * now;
now *= es[i];
}
}
void operator()(vm &a, bool inverse = false) {
int n = a.size();
int h = __builtin_ctz(n);
if (inverse) {
rrep(ph, h + 1, 1) {
int w = 1 << (ph - 1), p = 1 << (h - ph);
mint now = 1;
rep(s, w) {
int offset = s << (h - ph + 1);
rep(i, p) {
auto l = a[i + offset];
auto r = a[i + offset + p];
a[i + offset] = l + r;
a[i + offset + p] = (l - r) * now;
}
now *= sum_ie[__builtin_ctz(~(unsigned int) (s))];
}
}
mint iv = mint(n).inv();
rep(i, n) a[i] *= iv;
} else {
rep(ph, 1, h + 1) {
int w = 1 << (ph - 1), p = 1 << (h - ph);
mint now = 1;
rep(s, w) {
int offset = s << (h - ph + 1);
rep(i, p) {
auto l = a[i + offset];
auto r = a[i + offset + p] * now;
a[i + offset] = l + r;
a[i + offset + p] = l - r;
}
now *= sum_e[__builtin_ctz(~(unsigned int) (s))];
}
}
}
}
} ntt;
class fps : public vector<mint> {
static fps convolution(const fps &a, const fps &b) {
if (a.empty()) return {};
if (b.empty()) return {};
int s = a.size() + b.size() - 1;
if (min(a.size(), b.size()) <= 50) {
fps res(s);
if (a.size() >= b.size()) {
rep(i, a.size()) rep(j, b.size()) res[i + j] += a[i] * b[j];
} else {
rep(j, b.size()) rep(i, a.size()) res[i + j] += a[i] * b[j];
}
return res;
}
int t = 1;
while (t < s) t *= 2;
fps A(t), B(t);
rep(i, a.size()) A[i] = a[i];
rep(i, b.size()) B[i] = b[i];
ntt(A);
ntt(B);
rep(i, t) A[i] *= B[i];
ntt(A, true);
A.resize(s);
return A;
}
public:
using vector<mint>::vector;
mint eval(mint x) const {
mint res = 0;
mint now = 1;
rep(i, this->size()) {
res += (*this)[i] * now;
now *= x;
}
return res;
}
fps pre(int n) const {
return fps(this->begin(), this->begin() + min(n, (int) this->size()));
}
// return f'(x)
fps differ() const {
int n = this->size();
fps res(n - 1);
rep2(i, 1, n) res[i - 1] = (*this)[i] * i;
return res;
}
// return ∫ f(x)dx
fps integral() const {
int n = this->size();
if (n == 0) return fps();
fps res(n + 1);
rep(i, n) res[i + 1] = (*this)[i] / (i + 1);
return res;
};
fps operator>>(int n) const {
if ((int) this->size() <= n) return {};
fps res(*this);
res.erase(res.begin(), res.begin() + n);
return res;
}
fps operator<<(int n) const {
fps res(*this);
res.insert(res.begin(), n, 0);
return res;
}
fps &operator+=(const fps &a) {
if (this->size() < a.size()) this->resize(a.size());
rep(i, a.size()) (*this)[i] += a[i];
return *this;
}
fps &operator-=(const fps &a) {
if (this->size() < a.size()) this->resize(a.size());
rep(i, a.size()) (*this)[i] -= a[i];
return *this;
}
fps &operator*=(const fps &a) {
return *this = fps::convolution(*this, a);
}
fps &operator*=(mint k) {
rep(i, this->size()) (*this)[i] *= k;
return *this;
}
fps operator+(const fps &a) const {
fps res(*this);
return res += a;
}
fps operator-(const fps &a) const {
fps res(*this);
return res -= a;
}
fps operator*(const fps &a) const {
fps res(*this);
return res *= a;
}
fps operator*(mint k) const {
fps res(*this);
return res *= k;
}
// // P /= (ax + b)
// constexpr void divide(T a = 0, T b = 1) {
// int n = this->size();
// assert(n >= 2);
// assert(a != 0 or b != 0);
// if (b == T(0)) {
// assert((*this)[0] == T(0));
// T inv = T(1) / a;
// rep(i, n - 1) (*this)[i] = (*this)[i + 1] * inv;
// this->back() = T(0);
// } else {
// T inv = T(1) / b;
// rep(i, n - 1) {
// (*this)[i] *= inv;
// (*this)[i + 1] -= (*this)[i] * a;
// }
// assert(this->back() == T(0));
// }
// }
// reference of inv, log, exp, pow : https://opt-cp.com/fps-fast-algorithms/
// time complexity : O(n log n)
fps inv(int deg = -1) const {
int n = this->size();
assert(n and (*this)[0].val());
if (deg == -1) deg = n;
fps res(deg);
res[0] = (*this)[0].inv();
for (int m = 1; m < deg; m <<= 1) {
fps f(2 * m), g(2 * m);
rep(i, min(n, 2 * m)) f[i] = (*this)[i];
rep(i, m) g[i] = res[i];
ntt(f), ntt(g);
rep(i, 2 * m) f[i] *= g[i];
ntt(f, true);
rep(i, m) f[i] = 0;
ntt(f);
rep(i, 2 * m) f[i] *= g[i];
ntt(f, true);
rep(i, m, min(2 * m, deg)) res[i] = -f[i];
}
return res;
}
fps ÷_inplace(const fps &a, int d = -1) {
int n = this->size();
if (d == -1) d = n;
assert(d >= 0);
*this = convolution(*this, a.inv(d));
this->resize(d);
return *this;
}
fps divide(const fps &a, int d = -1) {
fps res(*this);
return res.divide_inplace(a, d);
}
// time complexity : O(n log n)
fps log(int deg = -1) const {
int n = this->size();
assert(n and (*this)[0].val() == 1);
if (deg == -1) deg = n;
fps res(this->differ());
res.divide_inplace(*this, deg);
res = res.integral();
res.pop_back();
return res;
}
// time complexity : O(n log n)
fps exp(int deg = -1) const {
int n = this->size();
assert(n and (*this)[0].val() == 0);
if (deg == -1) deg = n;
fps g{1}, g_fft, f(*this);
f.resize(deg);
f[0] = 1;
fps h_prime(this->differ());
h_prime.pb(0);
for (int m = 1; m < deg; m *= 2) {
// prepare
fps f_fft(f.begin(), f.begin() + m);
f_fft.resize(2 * m);
ntt(f_fft);
// Step 2.a'
if (m > 1) {
fps _f(m);
rep(i, m) _f[i] = f_fft[i] * g_fft[i];
ntt(_f, true);
_f.erase(_f.begin(), _f.begin() + m / 2);
_f.resize(m), ntt(_f);
rep(i, m) _f[i] *= g_fft[i];
ntt(_f, true);
_f.resize(m / 2);
_f *= -1;
g.insert(g.end(), _f.begin(), _f.begin() + m / 2);
}
// Step 2.b'--d'
fps t(f.begin(), f.begin() + m);
t = t.differ();
t.pb(0);
{
// Step 2.b'
fps r(h_prime.begin(), h_prime.begin() + m - 1);
// Step 2.c'
r.resize(m);
ntt(r);
rep(i, m) r[i] *= f_fft[i];
ntt(r, true);
// Step 2.d'
t -= r;
t.insert(t.begin(), t.back());
t.pop_back();
}
// Step 2.e'
t.resize(2 * m);
ntt(t);
g_fft = g;
g_fft.resize(2 * m);
ntt(g_fft);
rep(i, 2 * m) t[i] *= g_fft[i];
ntt(t, true);
t.resize(m);
// Step 2.f'
fps v(f.begin() + m, f.begin() + min(deg, 2 * m));
v.resize(m);
t.insert(t.begin(), m - 1, 0);
t.push_back(0);
t = t.integral();
rep(i, m) v[i] -= t[m + i];
// Step 2.g'
v.resize(2 * m);
ntt(v);
rep(i, 2 * m) v[i] *= f_fft[i];
ntt(v, true);
v.resize(m);
// Step 2.h'
rep(i, min(deg - m, m)) f[m + i] = v[i];
}
return f;
}
// time complexity : O(n log n)
fps pow(ll k, int deg = -1) const {
int n = this->size();
if (deg == -1) deg = n;
assert(k >= 0);
if (k == 0) {
fps res(deg);
if (deg > 0) res[0] = 1;
return res;
}
int l = 0;
while (l < n && (*this)[l].val() == 0) ++l;
if (l > (deg - 1) / k or l == n) return fps(deg);
mint c = (*this)[l];
fps res(this->begin() + l, this->end());
res *= c.inv();
res = res.log(deg - l * k);
res *= k;
res = res.exp();
res *= c.pow(k);
res.insert(res.begin(), l * k, 0);
return res;
}
// time complexity : O(nt) where t is the number of non-zero elements
fps sparse_pow(ll k, int deg = -1) const {
int n = this->size();
if (deg == -1) deg = n;
assert(k >= 0);
if (deg == 0) return {};
if (k == 0) {
fps res(deg);
res[0] = 1;
return res;
}
int l = 0;
while (l < n && (*this)[l].val() == 0) ++l;
if (l > (deg - 1) / k or l == n) return fps(deg);
deg -= l * k;
vector<pair<int, mint>> v;
rep(i, n) if ((*this)[i].val()) v.eb(i - l, (*this)[i]);
fps res(deg);
res[0] = v[0].second.pow(k);
mint iv_v0 = v[0].second.inv();
vm iv(deg, 1);
rep(i, 1, deg) {
// g = f^k
// g'f = kgf'
for (auto [d, coef]: v) {
if (!d) continue;
if (d > i) break;
res[i] += coef * d * res[i - d];
}
res[i] *= k;
for (auto [d, coef]: v) {
if (!d) continue;
if (d >= i) break;
res[i] -= coef * res[i - d] * (i - d);
}
res[i] *= iv_v0 * iv[i];
if (i + 1 < deg) iv[i + 1] = -iv[mint::get_mod() % (i + 1)] * (mint::get_mod() / (i + 1));
}
res.insert(res.begin(), l * k, 0);
return res;
}
// constraints : ∃t, t^2 ≡ f_s and s is even
// where s is the smallest index s.t. f_s != 0
// time complexity : O(n log n)
// reference : https://nyaannyaan.github.io/library/fps/fps-sqrt.hpp.html
fps sqrt(int deg = -1) const {
int n = this->size();
if (deg == -1) deg = n;
if (!n) return fps(deg);
if ((*this)[0] == mint(0)) {
rep(i, 1, n) {
if ((*this)[i] != mint(0)) {
assert(~i & 1);
if (deg - i / 2 <= 0) break;
fps res = ((*this) >> i).sqrt(deg - i / 2);
res = res << (i / 2);
assert((int) res.size() == deg);
return res;
}
}
return fps(deg, 0);
}
mint sqr = (*this)[0].sqrt();
assert(sqr * sqr == (*this)[0]);
fps res = {sqr};
mint inv2 = mint(2).inv();
for (int i = 1; i < deg; i <<= 1) {
res = (res + this->pre(i << 1) * res.inv(i << 1)) * inv2;
}
return res.pre(deg);
}
// calc f(x + c)
// time complexity : O(n log n)
fps taylor_shift(mint c) const {
int n = this->size();
vm fact(n), ifact(n);
fact[0] = 1;
rep(i, 1, n) fact[i] = fact[i - 1] * i;
ifact[n - 1] = fact[n - 1].inv();
rrep(i, n - 1) ifact[i] = ifact[i + 1] * (i + 1);
fps f(n), g(n);
mint nc = 1;
rep(i, n) {
f[i] = (*this)[n - 1 - i] * fact[n - 1 - i];
g[i] = nc * ifact[i];
nc *= c;
}
fps h = f * g;
fps res(n);
rep(i, n) res[i] = ifact[i] * h[n - 1 - i];
return res;
}
};
class combination {
public:
vector<mint> fact, ifact;
combination(int n) : fact(n + 1), ifact(n + 1) {
fact[0] = 1;
for (int i = 1; i <= n; ++i) fact[i] = fact[i - 1] * i;
ifact[n] = fact[n].inv();
for (int i = n; i >= 1; --i) ifact[i - 1] = ifact[i] * i;
}
mint operator()(int n, int k) {
if (k < 0 || k > n) return 0;
return fact[n] * ifact[k] * ifact[n - k];
}
} binom(2000000);
int main() {
INT(n, m);
vi b(n);
scan(b);
vi v(n - 1), mn_x(n - 1), mn_y(n - 1);
rep(i, n - 1) {
v[i] = b[i + 1] - b[i]; // type 2 - type 1 at i
mn_x[i] = (v[i] >= 0 ? 0 : -v[i]);
mn_y[i] = mn_x[i] + v[i];
}
int x = SUM(mn_x);
int y = SUM(mn_y);
if (x + y > m or x > b[0]) fin(0);
int rem = b[0] - x;
assert(b[n - 1] - y == rem);
if (x + y + rem > m) fin(0);
debug(x, y, rem);
debug(mn_x);
debug(mn_y);
fps f(rem + 1);
f[0] = 1;
rep(i, n - 1) {
fps g(rem + 1);
rep(j, rem + 1) g[j] = binom.ifact[mn_x[i] + j] * binom.ifact[mn_y[i] + j];
f *= g;
f.resize(rem + 1);
}
mint sum;
rep(i, rem + 1) {
if (x + y + rem + i > m) break;
mint ans = f[i];
debug(ans);
ans *= binom(m, x + y + rem + i);
ans *= binom(x + y + rem + i, x + y + 2 * i);
ans *= binom(x + y + 2 * i, x + i);
ans *= binom.fact[x + i];
ans *= binom.fact[y + i];
sum += ans;
debug(i, ans);
}
fin(sum);
}