#include using namespace std; #define FOR(i,m,n) for(int i=(m);i<(n);++i) #define REP(i,n) FOR(i,0,n) using ll = long long; constexpr int INF = 0x3f3f3f3f; constexpr long long LINF = 0x3f3f3f3f3f3f3f3fLL; constexpr double EPS = 1e-8; constexpr int MOD = 998244353; // constexpr int MOD = 1000000007; constexpr int DY4[]{1, 0, -1, 0}, DX4[]{0, -1, 0, 1}; constexpr int DY8[]{1, 1, 0, -1, -1, -1, 0, 1}; constexpr int DX8[]{0, -1, -1, -1, 0, 1, 1, 1}; template inline bool chmax(T& a, U b) { return a < b ? (a = b, true) : false; } template inline bool chmin(T& a, U b) { return a > b ? (a = b, true) : false; } struct IOSetup { IOSetup() { std::cin.tie(nullptr); std::ios_base::sync_with_stdio(false); std::cout << fixed << setprecision(20); } } iosetup; template struct MInt { unsigned int v; constexpr MInt() : v(0) {} constexpr MInt(const long long x) : v(x >= 0 ? x % M : x % M + M) {} static constexpr MInt raw(const int x) { MInt x_; x_.v = x; return x_; } static constexpr int get_mod() { return M; } static constexpr void set_mod(const int divisor) { assert(std::cmp_equal(divisor, M)); } static void init(const int x) { inv(x); fact(x); fact_inv(x); } template static MInt inv(const int n) { // assert(0 <= n && n < M && std::gcd(n, M) == 1); static std::vector inverse{0, 1}; const int prev = inverse.size(); if (n < prev) return inverse[n]; if constexpr (MEMOIZES) { // "n!" and "M" must be disjoint. inverse.resize(n + 1); for (int i = prev; i <= n; ++i) { inverse[i] = -inverse[M % i] * raw(M / i); } return inverse[n]; } int u = 1, v = 0; for (unsigned int a = n, b = M; b;) { const unsigned int q = a / b; std::swap(a -= q * b, b); std::swap(u -= q * v, v); } return u; } static MInt fact(const int n) { static std::vector factorial{1}; if (const int prev = factorial.size(); n >= prev) { factorial.resize(n + 1); for (int i = prev; i <= n; ++i) { factorial[i] = factorial[i - 1] * i; } } return factorial[n]; } static MInt fact_inv(const int n) { static std::vector f_inv{1}; if (const int prev = f_inv.size(); n >= prev) { f_inv.resize(n + 1); f_inv[n] = inv(fact(n).v); for (int i = n; i > prev; --i) { f_inv[i - 1] = f_inv[i] * i; } } return f_inv[n]; } static MInt nCk(const int n, const int k) { if (n < 0 || n < k || k < 0) [[unlikely]] return MInt(); return fact(n) * (n - k < k ? fact_inv(k) * fact_inv(n - k) : fact_inv(n - k) * fact_inv(k)); } static MInt nPk(const int n, const int k) { return n < 0 || n < k || k < 0 ? MInt() : fact(n) * fact_inv(n - k); } static MInt nHk(const int n, const int k) { return n < 0 || k < 0 ? MInt() : (k == 0 ? 1 : nCk(n + k - 1, k)); } static MInt large_nCk(long long n, const int k) { if (n < 0 || n < k || k < 0) [[unlikely]] return MInt(); inv(k); MInt res = 1; for (int i = 1; i <= k; ++i) { res *= inv(i) * n--; } return res; } constexpr MInt pow(long long exponent) const { MInt res = 1, tmp = *this; for (; exponent > 0; exponent >>= 1) { if (exponent & 1) res *= tmp; tmp *= tmp; } return res; } constexpr MInt& operator+=(const MInt& x) { if ((v += x.v) >= M) v -= M; return *this; } constexpr MInt& operator-=(const MInt& x) { if ((v += M - x.v) >= M) v -= M; return *this; } constexpr MInt& operator*=(const MInt& x) { v = (unsigned long long){v} * x.v % M; return *this; } MInt& operator/=(const MInt& x) { return *this *= inv(x.v); } constexpr auto operator<=>(const MInt& x) const = default; constexpr MInt& operator++() { if (++v == M) [[unlikely]] v = 0; return *this; } constexpr MInt operator++(int) { const MInt res = *this; ++*this; return res; } constexpr MInt& operator--() { v = (v == 0 ? M - 1 : v - 1); return *this; } constexpr MInt operator--(int) { const MInt res = *this; --*this; return res; } constexpr MInt operator+() const { return *this; } constexpr MInt operator-() const { return raw(v ? M - v : 0); } constexpr MInt operator+(const MInt& x) const { return MInt(*this) += x; } constexpr MInt operator-(const MInt& x) const { return MInt(*this) -= x; } constexpr MInt operator*(const MInt& x) const { return MInt(*this) *= x; } MInt operator/(const MInt& x) const { return MInt(*this) /= x; } friend std::ostream& operator<<(std::ostream& os, const MInt& x) { return os << x.v; } friend std::istream& operator>>(std::istream& is, MInt& x) { long long v; is >> v; x = MInt(v); return is; } }; using ModInt = MInt; #include template struct NumberTheoreticTransform { using ModInt = MInt; NumberTheoreticTransform() = default; template std::vector dft(const std::vector& a); void idft(std::vector* a); template std::vector convolution( const std::vector& a, const std::vector& b) const { const int a_size = a.size(), b_size = b.size(); std::vector> c(a_size), d(b_size); for (int i = 0; i < a_size; ++i) { c[i] = atcoder::static_modint::raw(ModInt(a[i]).v); } for (int i = 0; i < b_size; ++i) { d[i] = atcoder::static_modint::raw(ModInt(b[i]).v); } c = atcoder::convolution(c, d); const int c_size = c.size(); std::vector res(c_size); for (int i = 0; i < c_size; ++i) { res[i] = ModInt::raw(c[i].val()); } return res; } }; template struct FormalPowerSeries { std::vector coef; explicit FormalPowerSeries(const int deg = 0) : coef(deg + 1, 0) {} explicit FormalPowerSeries(const std::vector& coef) : coef(coef) {} FormalPowerSeries(const std::initializer_list init) : coef(init.begin(), init.end()) {} template explicit FormalPowerSeries(const InputIter first, const InputIter last) : coef(first, last) {} inline const T& operator[](const int term) const { return coef[term]; } inline T& operator[](const int term) { return coef[term]; } using Mult = std::function(const std::vector&, const std::vector&)>; using Sqrt = std::function; static void set_mult(const Mult mult) { get_mult() = mult; } static void set_sqrt(const Sqrt sqrt) { get_sqrt() = sqrt; } void resize(const int deg) { coef.resize(deg + 1, 0); } void shrink() { while (coef.size() > 1 && coef.back() == 0) coef.pop_back(); } int degree() const { return std::ssize(coef) - 1; } FormalPowerSeries& operator=(const std::vector& coef_) { coef = coef_; return *this; } FormalPowerSeries& operator=(const FormalPowerSeries& x) = default; FormalPowerSeries& operator+=(const FormalPowerSeries& x) { const int deg_x = x.degree(); if (deg_x > degree()) resize(deg_x); for (int i = 0; i <= deg_x; ++i) { coef[i] += x[i]; } return *this; } FormalPowerSeries& operator-=(const FormalPowerSeries& x) { const int deg_x = x.degree(); if (deg_x > degree()) resize(deg_x); for (int i = 0; i <= deg_x; ++i) { coef[i] -= x[i]; } return *this; } FormalPowerSeries& operator*=(const T x) { for (T& e : coef) e *= x; return *this; } FormalPowerSeries& operator*=(const FormalPowerSeries& x) { return *this = get_mult()(coef, x.coef); } FormalPowerSeries& operator/=(const T x) { assert(x != 0); return *this *= static_cast(1) / x; } FormalPowerSeries& operator/=(const FormalPowerSeries& x) { const int n = degree() - x.degree() + 1; if (n <= 0) return *this = FormalPowerSeries(); const std::vector tmp = get_mult()( std::vector(coef.rbegin(), std::next(coef.rbegin(), n)), FormalPowerSeries( x.coef.rbegin(), std::next(x.coef.rbegin(), std::min(x.degree() + 1, n))) .inv(n - 1).coef); return *this = FormalPowerSeries(std::prev(tmp.rend(), n), tmp.rend()); } FormalPowerSeries& operator%=(const FormalPowerSeries& x) { if (x.degree() == 0) return *this = FormalPowerSeries{0}; *this -= *this / x * x; resize(x.degree() - 1); return *this; } FormalPowerSeries& operator<<=(const int n) { coef.insert(coef.begin(), n, 0); return *this; } FormalPowerSeries& operator>>=(const int n) { if (degree() < n) return *this = FormalPowerSeries(); coef.erase(coef.begin(), coef.begin() + n); return *this; } bool operator==(FormalPowerSeries x) const { x.shrink(); FormalPowerSeries y = *this; y.shrink(); return x.coef == y.coef; } FormalPowerSeries operator+() const { return *this; } FormalPowerSeries operator-() const { FormalPowerSeries res = *this; for (T& e : res.coef) e = -e; return res; } FormalPowerSeries operator+(const FormalPowerSeries& x) const { return FormalPowerSeries(*this) += x; } FormalPowerSeries operator-(const FormalPowerSeries& x) const { return FormalPowerSeries(*this) -= x; } FormalPowerSeries operator*(const T x) const { return FormalPowerSeries(*this) *= x; } FormalPowerSeries operator*(const FormalPowerSeries& x) const { return FormalPowerSeries(*this) *= x; } FormalPowerSeries operator/(const T x) const { return FormalPowerSeries(*this) /= x; } FormalPowerSeries operator/(const FormalPowerSeries& x) const { return FormalPowerSeries(*this) /= x; } FormalPowerSeries operator%(const FormalPowerSeries& x) const { return FormalPowerSeries(*this) %= x; } FormalPowerSeries operator<<(const int n) const { return FormalPowerSeries(*this) <<= n; } FormalPowerSeries operator>>(const int n) const { return FormalPowerSeries(*this) >>= n; } T horner(const T x) const { return std::accumulate( coef.rbegin(), coef.rend(), static_cast(0), [x](const T l, const T r) -> T { return l * x + r; }); } FormalPowerSeries differential() const { const int deg = degree(); assert(deg >= 0); FormalPowerSeries res(std::max(deg - 1, 0)); for (int i = 1; i <= deg; ++i) { res[i - 1] = coef[i] * i; } return res; } FormalPowerSeries exp(const int deg) const { assert(coef[0] == 0); const int n = coef.size(); const FormalPowerSeries one{1}; FormalPowerSeries res = one; for (int i = 1; i <= deg; i <<= 1) { res *= FormalPowerSeries(coef.begin(), std::next(coef.begin(), std::min(n, i << 1))) - res.log((i << 1) - 1) + one; res.coef.resize(i << 1); } res.resize(deg); return res; } FormalPowerSeries exp() const { return exp(degree()); } FormalPowerSeries inv(const int deg) const { assert(coef[0] != 0); const int n = coef.size(); FormalPowerSeries res{static_cast(1) / coef[0]}; for (int i = 1; i <= deg; i <<= 1) { res = res + res - res * res * FormalPowerSeries( coef.begin(), std::next(coef.begin(), std::min(n, i << 1))); res.coef.resize(i << 1); } res.resize(deg); return res; } FormalPowerSeries inv() const { return inv(degree()); } FormalPowerSeries log(const int deg) const { assert(coef[0] == 1); FormalPowerSeries integrand = differential() * inv(deg - 1); integrand.resize(deg); for (int i = deg; i > 0; --i) { integrand[i] = integrand[i - 1] / i; } integrand[0] = 0; return integrand; } FormalPowerSeries log() const { return log(degree()); } FormalPowerSeries pow(long long exponent, const int deg) const { const int n = coef.size(); if (exponent == 0) { FormalPowerSeries res(deg); if (deg != -1) [[unlikely]] res[0] = 1; return res; } assert(deg >= 0); for (int i = 0; i < n; ++i) { if (coef[i] == 0) continue; if (i > deg / exponent) break; const long long shift = exponent * i; T tmp = 1, base = coef[i]; for (long long e = exponent; e > 0; e >>= 1) { if (e & 1) tmp *= base; base *= base; } const FormalPowerSeries res = ((*this >> i) / coef[i]).log(deg - shift); return ((res * exponent).exp(deg - shift) * tmp) << shift; } return FormalPowerSeries(deg); } FormalPowerSeries pow(const long long exponent) const { return pow(exponent, degree()); } FormalPowerSeries mod_pow(long long exponent, const FormalPowerSeries& md) const { const int deg = md.degree() - 1; if (deg < 0) [[unlikely]] return FormalPowerSeries(-1); const FormalPowerSeries inv_rev_md = FormalPowerSeries(md.coef.rbegin(), md.coef.rend()).inv(); const auto mod_mult = [&md, &inv_rev_md, deg]( FormalPowerSeries* multiplicand, const FormalPowerSeries& multiplier) -> void { *multiplicand *= multiplier; if (deg < multiplicand->degree()) { const int n = multiplicand->degree() - deg; const FormalPowerSeries quotient = FormalPowerSeries(multiplicand->coef.rbegin(), std::next(multiplicand->coef.rbegin(), n)) * FormalPowerSeries( inv_rev_md.coef.begin(), std::next(inv_rev_md.coef.begin(), std::min(deg + 2, n))); *multiplicand -= FormalPowerSeries(std::prev(quotient.coef.rend(), n), quotient.coef.rend()) * md; multiplicand->resize(deg); } multiplicand->shrink(); }; FormalPowerSeries res{1}, base = *this; for (; exponent > 0; exponent >>= 1) { if (exponent & 1) mod_mult(&res, base); mod_mult(&base, base); } return res; } FormalPowerSeries sqrt(const int deg) const { const int n = coef.size(); if (coef[0] == 0) { for (int i = 1; i < n; ++i) { if (coef[i] == 0) continue; if (i & 1) return FormalPowerSeries(-1); const int shift = i >> 1; if (deg < shift) break; FormalPowerSeries res = (*this >> i).sqrt(deg - shift); if (res.coef.empty()) return FormalPowerSeries(-1); res <<= shift; res.resize(deg); return res; } return FormalPowerSeries(deg); } T s; if (!get_sqrt()(coef.front(), &s)) return FormalPowerSeries(-1); FormalPowerSeries res{s}; const T half = static_cast(1) / 2; for (int i = 1; i <= deg; i <<= 1) { res = (FormalPowerSeries(coef.begin(), std::next(coef.begin(), std::min(n, i << 1))) * res.inv((i << 1) - 1) + res) * half; } res.resize(deg); return res; } FormalPowerSeries sqrt() const { return sqrt(degree()); } FormalPowerSeries translate(const T c) const { const int n = coef.size(); std::vector fact(n, 1), inv_fact(n, 1); for (int i = 1; i < n; ++i) { fact[i] = fact[i - 1] * i; } inv_fact[n - 1] = static_cast(1) / fact[n - 1]; for (int i = n - 1; i > 0; --i) { inv_fact[i - 1] = inv_fact[i] * i; } std::vector g(n), ex(n); for (int i = 0; i < n; ++i) { g[i] = coef[i] * fact[i]; } std::reverse(g.begin(), g.end()); T pow_c = 1; for (int i = 0; i < n; ++i) { ex[i] = pow_c * inv_fact[i]; pow_c *= c; } const std::vector conv = get_mult()(g, ex); FormalPowerSeries res(n - 1); for (int i = 0; i < n; ++i) { res[i] = conv[n - 1 - i] * inv_fact[i]; } return res; } private: static Mult& get_mult() { static Mult mult = [](const std::vector& a, const std::vector& b) -> std::vector { const int n = a.size(), m = b.size(); std::vector res(n + m - 1, 0); for (int i = 0; i < n; ++i) { for (int j = 0; j < m; ++j) { res[i + j] += a[i] * b[j]; } } return res; }; return mult; } static Sqrt& get_sqrt() { static Sqrt sqrt = [](const T&, T*) -> bool { return false; }; return sqrt; } }; int main() { FormalPowerSeries::set_mult( [](const vector& a, const vector& b) -> vector { static NumberTheoreticTransform ntt; return ntt.convolution(a, b); }); int n, m; cin >> n >> m; vector b(n); for (int& b_i : b) cin >> b_i; vector c(n); c.front() = b.front() - (m - b.back()); FOR(i, 1, n) c[i] = b[i] - b[i - 1]; // REP(i, n) cout << c[i] << " \n"[i + 1 == n]; FormalPowerSeries f{1}; REP(i, n) { FormalPowerSeries g(m); for (int j = 0; j * 2 + abs(c[i]) <= m; ++j) { g[j * 2 + abs(c[i])] = ModInt::fact_inv(j) * ModInt::fact_inv(j + abs(c[i])); } f *= g; if (f.degree() > m) f.resize(m); } cout << f[m] * ModInt::fact(m) << '\n'; return 0; }