#define PROBLEM "https://yukicoder.me/problems/no/2005"

#include <iostream>
#include <atcoder/convolution>
#include <atcoder/modint>

using mint = atcoder::modint998244353;

std::istream& operator>>(std::istream& in, mint &a) {
    long long e; in >> e; a = e;
    return in;
}

std::ostream& operator<<(std::ostream& out, const mint &a) {
    out << a.val();
    return out;
}

#include <algorithm>
#include <cassert>
#include <iostream>
#include <queue>

#include <cmath>
#include <limits>
#include <type_traits>
#include <vector>

namespace suisen {
// ! utility
template <typename ...Types>
using constraints_t = std::enable_if_t<std::conjunction_v<Types...>, std::nullptr_t>;
template <bool cond_v, typename Then, typename OrElse>
constexpr decltype(auto) constexpr_if(Then&& then, OrElse&& or_else) {
    if constexpr (cond_v) {
        return std::forward<Then>(then);
    } else {
        return std::forward<OrElse>(or_else);
    }
}

// ! function
template <typename ReturnType, typename Callable, typename ...Args>
using is_same_as_invoke_result = std::is_same<std::invoke_result_t<Callable, Args...>, ReturnType>;
template <typename F, typename T>
using is_uni_op = is_same_as_invoke_result<T, F, T>;
template <typename F, typename T>
using is_bin_op = is_same_as_invoke_result<T, F, T, T>;

template <typename Comparator, typename T>
using is_comparator = std::is_same<std::invoke_result_t<Comparator, T, T>, bool>;

// ! integral
template <typename T, typename = constraints_t<std::is_integral<T>>>
constexpr int bit_num = std::numeric_limits<std::make_unsigned_t<T>>::digits;
template <typename T, unsigned int n>
struct is_nbit { static constexpr bool value = bit_num<T> == n; };
template <typename T, unsigned int n>
static constexpr bool is_nbit_v = is_nbit<T, n>::value;

// ?
template <typename T>
struct safely_multipliable {};
template <>
struct safely_multipliable<int> { using type = long long; };
template <>
struct safely_multipliable<long long> { using type = __int128_t; };
template <>
struct safely_multipliable<unsigned int> { using type = unsigned long long; };
template <>
struct safely_multipliable<unsigned long int> { using type = __uint128_t; };
template <>
struct safely_multipliable<unsigned long long> { using type = __uint128_t; };
template <>
struct safely_multipliable<float> { using type = float; };
template <>
struct safely_multipliable<double> { using type = double; };
template <>
struct safely_multipliable<long double> { using type = long double; };
template <typename T>
using safely_multipliable_t = typename safely_multipliable<T>::type;

template <typename T, typename = void>
struct rec_value_type {
    using type = T;
};
template <typename T>
struct rec_value_type<T, std::void_t<typename T::value_type>> {
    using type = typename rec_value_type<typename T::value_type>::type;
};
template <typename T>
using rec_value_type_t = typename rec_value_type<T>::type;

} // namespace suisen

#include <optional>

/**
 * refernce: https://37zigen.com/tonelli-shanks-algorithm/
 * calculates x s.t. x^2 = a mod p in O((log p)^2).
 */
template <typename mint>
std::optional<mint> safe_sqrt(mint a) {
    static int p = mint::mod();
    if (a == 0) return std::make_optional(0);
    if (p == 2) return std::make_optional(a);
    if (a.pow((p - 1) / 2) != 1) return std::nullopt;
    mint b = 1;
    while (b.pow((p - 1) / 2) == 1) ++b;
    static int tlz = __builtin_ctz(p - 1), q = (p - 1) >> tlz;
    mint x = a.pow((q + 1) / 2);
    b = b.pow(q);
    for (int shift = 2; x * x != a; ++shift) {
        mint e = a.inv() * x * x;
        if (e.pow(1 << (tlz - shift)) != 1) x *= b;
        b *= b;
    }
    return std::make_optional(x);
}

/**
 * calculates x s.t. x^2 = a mod p in O((log p)^2).
 * if not exists, raises runtime error.
 */
template <typename mint>
auto sqrt(mint a) -> decltype(mint::mod(), mint()) {
    return *safe_sqrt(a);
}
template <typename mint>
auto log(mint a) -> decltype(mint::mod(), mint()) {
    assert(a == 1);
    return 0;
}
template <typename mint>
auto exp(mint a) -> decltype(mint::mod(), mint()) {
    assert(a == 0);
    return 1;
}
template <typename mint, typename T>
auto pow(mint a, T b) -> decltype(mint::mod(), mint()) {
    return a.pow(b);
}
template <typename mint>
auto inv(mint a) -> decltype(mint::mod(), mint()) {
    return a.inv();
}

namespace suisen {
template <typename mint>
class inv_mods {
    public:
        inv_mods() {}
        inv_mods(int n) { ensure(n); }
        const mint& operator[](int i) const {
            ensure(i);
            return invs[i];
        }
        static void ensure(int n) {
            int sz = invs.size();
            if (sz < 2) invs = {0, 1}, sz = 2;
            if (sz < n + 1) {
                invs.resize(n + 1);
                for (int i = sz; i <= n; ++i) invs[i] = mint(mod - mod / i) * invs[mod % i];
            }
        }
    private:
        static std::vector<mint> invs;
        static constexpr int mod = mint::mod();
};
template <typename mint>
std::vector<mint> inv_mods<mint>::invs{};
}

namespace suisen {
    template <typename T>
    struct FPSNaive : std::vector<T> {
        static inline int MAX_DEG = std::numeric_limits<int>::max() / 2;

        using value_type = T;
        using element_type = rec_value_type_t<T>;
        using std::vector<value_type>::vector;

        FPSNaive(const std::initializer_list<value_type> l) : std::vector<value_type>::vector(l) {}
        FPSNaive(const std::vector<value_type>& v) : std::vector<value_type>::vector(v) {}

        static void set_max_deg(int max_deg) {
            FPSNaive<T>::MAX_DEG = max_deg;
        }

        const value_type operator[](int n) const {
            return n <= deg() ? unsafe_get(n) : value_type{ 0 };
        }
        value_type& operator[](int n) {
            return ensure_deg(n), unsafe_get(n);
        }

        int size() const {
            return std::vector<value_type>::size();
        }
        int deg() const {
            return size() - 1;
        }
        int normalize() {
            while (size() and this->back() == value_type{ 0 }) this->pop_back();
            return deg();
        }
        FPSNaive& cut_inplace(int max_deg) {
            if (deg() > max_deg) this->resize(std::max(0, max_deg + 1));
            return *this;
        }
        FPSNaive cut(int max_deg) const {
            return FPSNaive(*this).cut_inplace(max_deg);
        }

        FPSNaive operator+() const {
            return FPSNaive(*this);
        }
        FPSNaive operator-() const {
            FPSNaive f(*this);
            for (auto& e : f) e = -e;
            return f;
        }
        FPSNaive& operator++() { return ++(*this)[0], * this; }
        FPSNaive& operator--() { return --(*this)[0], * this; }
        FPSNaive& operator+=(const value_type x) { return (*this)[0] += x, *this; }
        FPSNaive& operator-=(const value_type x) { return (*this)[0] -= x, *this; }
        FPSNaive& operator+=(const FPSNaive& g) {
            ensure_deg(g.deg());
            for (int i = 0; i <= g.deg(); ++i) unsafe_get(i) += g.unsafe_get(i);
            return *this;
        }
        FPSNaive& operator-=(const FPSNaive& g) {
            ensure_deg(g.deg());
            for (int i = 0; i <= g.deg(); ++i) unsafe_get(i) -= g.unsafe_get(i);
            return *this;
        }
        FPSNaive& operator*=(const FPSNaive& g) { return *this = *this * g; }
        FPSNaive& operator*=(const value_type x) {
            for (auto& e : *this) e *= x;
            return *this;
        }
        FPSNaive& operator/=(const FPSNaive& g) { return *this = *this / g; }
        FPSNaive& operator%=(const FPSNaive& g) { return *this = *this % g; }
        FPSNaive& operator<<=(const int shamt) {
            this->insert(this->begin(), shamt, value_type{ 0 });
            return *this;
        }
        FPSNaive& operator>>=(const int shamt) {
            if (shamt > size()) this->clear();
            else this->erase(this->begin(), this->begin() + shamt);
            return *this;
        }

        friend FPSNaive operator+(FPSNaive f, const FPSNaive& g) { f += g; return f; }
        friend FPSNaive operator+(FPSNaive f, const value_type& x) { f += x; return f; }
        friend FPSNaive operator-(FPSNaive f, const FPSNaive& g) { f -= g; return f; }
        friend FPSNaive operator-(FPSNaive f, const value_type& x) { f -= x; return f; }
        friend FPSNaive operator*(const FPSNaive& f, const FPSNaive& g) {
            if (f.empty() or g.empty()) return FPSNaive{};
            const int n = f.size(), m = g.size();
            FPSNaive h(std::min(MAX_DEG + 1, n + m - 1));
            for (int i = 0; i < n; ++i) for (int j = 0; j < m; ++j) {
                if (i + j > MAX_DEG) break;
                h.unsafe_get(i + j) += f.unsafe_get(i) * g.unsafe_get(j);
            }
            return h;
        }
        friend FPSNaive operator*(FPSNaive f, const value_type& x) { f *= x; return f; }
        friend FPSNaive operator/(FPSNaive f, const FPSNaive& g) { return std::move(f.div_mod(g).first); }
        friend FPSNaive operator%(FPSNaive f, const FPSNaive& g) { return std::move(f.div_mod(g).second); }
        friend FPSNaive operator*(const value_type x, FPSNaive f) { f *= x; return f; }
        friend FPSNaive operator<<(FPSNaive f, const int shamt) { f <<= shamt; return f; }
        friend FPSNaive operator>>(FPSNaive f, const int shamt) { f >>= shamt; return f; }

        std::pair<FPSNaive, FPSNaive> div_mod(FPSNaive g) const {
            FPSNaive f = *this;
            const int fd = f.normalize(), gd = g.normalize();
            assert(gd >= 0);
            if (fd < gd) return { FPSNaive{}, f };
            if (gd == 0) return { f *= g.unsafe_get(0).inv(), FPSNaive{} };
            const int k = f.deg() - gd;
            value_type head_inv = g.unsafe_get(gd).inv();
            FPSNaive q(k + 1);
            for (int i = k; i >= 0; --i) {
                value_type div = f.unsafe_get(i + gd) * head_inv;
                q.unsafe_get(i) = div;
                for (int j = 0; j <= gd; ++j) f.unsafe_get(i + j) -= div * g.unsafe_get(j);
            }
            return { q, f.cut_inplace(gd - 1) };
        }

        friend bool operator==(const FPSNaive& f, const FPSNaive& g) {
            const int n = f.size(), m = g.size();
            if (n < m) return g == f;
            for (int i = 0; i < m; ++i) if (f.unsafe_get(i) != g.unsafe_get(i)) return false;
            for (int i = m; i < n; ++i) if (f.unsafe_get(i) != 0) return false;
            return true;
        }
        friend bool operator!=(const FPSNaive& f, const FPSNaive& g) {
            return not (f == g);
        }

        FPSNaive mul(const FPSNaive& g, int max_deg = -1) const {
            if (max_deg < 0) max_deg = deg();
            if (this->empty() or g.empty()) return FPSNaive{};
            const int n = size(), m = g.size();
            FPSNaive h(std::min(max_deg + 1, n + m - 1));
            for (int i = 0; i < n; ++i) for (int j = 0; j < m; ++j) {
                if (i + j > max_deg) break;
                h.unsafe_get(i + j) += unsafe_get(i) * g.unsafe_get(j);
            }
            return h;
        }
        FPSNaive diff() const {
            if (this->empty()) return {};
            FPSNaive g(size() - 1);
            for (int i = 1; i <= deg(); ++i) g.unsafe_get(i - 1) = unsafe_get(i) * i;
            return g;
        }
        FPSNaive intg() const {
            const int n = size();
            FPSNaive g(n + 1);
            for (int i = 0; i < n; ++i) g.unsafe_get(i + 1) = unsafe_get(i) * invs[i + 1];
            if (g.deg() > MAX_DEG) g.cut_inplace(MAX_DEG);
            return g;
        }
        FPSNaive inv(int max_deg = -1) const {
            if (max_deg < 0) max_deg = deg();
            FPSNaive g(max_deg + 1);
            const value_type inv_f0 = ::inv(unsafe_get(0));
            g.unsafe_get(0) = inv_f0;
            for (int i = 1; i <= max_deg; ++i) {
                for (int j = 1; j <= i; ++j) g.unsafe_get(i) -= g.unsafe_get(i - j) * (*this)[j];
                g.unsafe_get(i) *= inv_f0;
            }
            return g;
        }
        FPSNaive exp(int max_deg = -1) const {
            if (max_deg < 0) max_deg = deg();
            assert(unsafe_get(0) == value_type{ 0 });
            FPSNaive g(max_deg + 1);
            g.unsafe_get(0) = value_type{ 1 };
            for (int i = 1; i <= max_deg; ++i) {
                for (int j = 1; j <= i; ++j) g.unsafe_get(i) += j * g.unsafe_get(i - j) * (*this)[j];
                g.unsafe_get(i) *= invs[i];
            }
            return g;
        }
        FPSNaive log(int max_deg = -1) const {
            if (max_deg < 0) max_deg = deg();
            assert(unsafe_get(0) == value_type{ 1 });
            FPSNaive g(max_deg + 1);
            g.unsafe_get(0) = value_type{ 0 };
            for (int i = 1; i <= max_deg; ++i) {
                g.unsafe_get(i) = i * (*this)[i];
                for (int j = 1; j < i; ++j) g.unsafe_get(i) -= (i - j) * g.unsafe_get(i - j) * (*this)[j];
                g.unsafe_get(i) *= invs[i];
            }
            return g;
        }
        FPSNaive pow(const long long k, int max_deg = -1) const {
            if (max_deg < 0) max_deg = deg();
            if (k == 0) {
                FPSNaive res(max_deg + 1);
                res[0] = 1;
                return res;
            }
            int z = 0;
            while (z < size() and unsafe_get(z) == value_type{ 0 }) ++z;
            if (z == size() or z > max_deg / k) return FPSNaive(max_deg + 1, 0);
            const int d = max_deg - z * k;

            FPSNaive g(d + 1);
            const value_type inv_f0 = ::inv(unsafe_get(z));
            g.unsafe_get(0) = unsafe_get(z).pow(k);
            for (int i = 1; i <= d; ++i) {
                for (int j = 1; j <= i; ++j) g.unsafe_get(i) += (element_type{ k } *j - (i - j)) * g.unsafe_get(i - j) * (*this)[z + j];
                g.unsafe_get(i) *= inv_f0 * invs[i];
            }
            g <<= z * k;
            return g;
        }

        std::optional<FPSNaive> safe_sqrt(int max_deg = -1) const {
            if (max_deg < 0) max_deg = deg();
            int dl = 0;
            while (dl < size() and unsafe_get(dl) == value_type{ 0 }) ++dl;
            if (dl == size()) return FPSNaive(max_deg + 1, 0);
            if (dl & 1) return std::nullopt;

            const int d = max_deg - dl / 2;

            FPSNaive g(d + 1);
            auto opt_g0 = ::safe_sqrt((*this)[dl]);
            if (not opt_g0.has_value()) return std::nullopt;
            g.unsafe_get(0) = *opt_g0;
            value_type inv_2g0 = ::inv(2 * g.unsafe_get(0));
            for (int i = 1; i <= d; ++i) {
                g.unsafe_get(i) = (*this)[dl + i];
                for (int j = 1; j < i; ++j) g.unsafe_get(i) -= g.unsafe_get(j) * g.unsafe_get(i - j);
                g.unsafe_get(i) *= inv_2g0;
            }
            g <<= dl / 2;
            return g;
        }
        FPSNaive sqrt(int max_deg = -1) const {
            if (max_deg < 0) max_deg = deg();
            return *safe_sqrt(max_deg);
        }

        value_type eval(value_type x) const {
            value_type y = 0;
            for (int i = size() - 1; i >= 0; --i) y = y * x + unsafe_get(i);
            return y;
        }

    private:
        static inline inv_mods<element_type> invs;

        void ensure_deg(int d) {
            if (deg() < d) this->resize(d + 1, value_type{ 0 });
        }
        const value_type& unsafe_get(int i) const {
            return std::vector<value_type>::operator[](i);
        }
        value_type& unsafe_get(int i) {
            return std::vector<value_type>::operator[](i);
        }
    };
} // namespace suisen

template <typename mint>
auto sqrt(suisen::FPSNaive<mint> a) -> decltype(mint::mod(), suisen::FPSNaive<mint>{}) {
    return a.sqrt(suisen::FPSNaive<mint>::MAX_DEG == std::numeric_limits<int>::max() / 2 ? suisen::FPSNaive<mint>::MAX_DEG : a.deg());
}
template <typename mint>
auto log(suisen::FPSNaive<mint> a) -> decltype(mint::mod(), suisen::FPSNaive<mint>{}) {
    return a.log(suisen::FPSNaive<mint>::MAX_DEG == std::numeric_limits<int>::max() / 2 ? suisen::FPSNaive<mint>::MAX_DEG : a.deg());
}
template <typename mint>
auto exp(suisen::FPSNaive<mint> a) -> decltype(mint::mod(), suisen::FPSNaive<mint>{}) {
    return a.exp(suisen::FPSNaive<mint>::MAX_DEG == std::numeric_limits<int>::max() / 2 ? suisen::FPSNaive<mint>::MAX_DEG : a.deg());
}
template <typename mint, typename T>
auto pow(suisen::FPSNaive<mint> a, T b) -> decltype(mint::mod(), suisen::FPSNaive<mint>{}) {
    return a.pow(b, suisen::FPSNaive<mint>::MAX_DEG == std::numeric_limits<int>::max() / 2 ? suisen::FPSNaive<mint>::MAX_DEG : a.deg());
}
template <typename mint>
auto inv(suisen::FPSNaive<mint> a) -> decltype(mint::mod(), suisen::FPSNaive<mint>{}) {
    return a.inv(suisen::FPSNaive<mint>::MAX_DEG == std::numeric_limits<int>::max() / 2 ? suisen::FPSNaive<mint>::MAX_DEG : a.deg());
}

namespace suisen {
    template <typename mint>
    using convolution_t = std::vector<mint>(*)(const std::vector<mint>&, const std::vector<mint>&);

    template <typename mint>
    struct FPS : public std::vector<mint> {
        using base_type = std::vector<mint>;
        using value_type = typename base_type::value_type;
        using base_type::vector;

        FPS(const std::initializer_list<mint> l) : std::vector<mint>::vector(l) {}
        FPS(const std::vector<mint>& v) : std::vector<mint>::vector(v) {}
        FPS(std::vector<mint>&& v) : std::vector<mint>::vector(std::move(v)) {}

        static void set_multiplication(convolution_t<mint> multiplication) {
            FPS<mint>::mult = multiplication;
        }

        int size() const noexcept {
            return base_type::size();
        }
        int deg() const noexcept {
            return size() - 1;
        }
        void ensure(int n) {
            if (size() < n) this->resize(n);
        }

        value_type safe_get(int d) const {
            return d <= deg() ? (*this)[d] : 0;
        }
        value_type& safe_get(int d) {
            ensure(d + 1);
            return (*this)[d];
        }

        int cut_trailing_zeros() {
            while (this->size() and this->back() == 0) this->pop_back();
            return deg();
        }
        void cut(int n) {
            if (size() > n) this->resize(std::max(0, n));
        }
        FPS cut_copy(int n) const {
            FPS res(this->begin(), this->begin() + std::min(size(), n));
            res.ensure(n);
            return res;
        }

        /* Unary Operations */

        FPS operator+() const { return *this; }
        FPS operator-() const {
            FPS res = *this;
            for (auto& e : res) e = -e;
            return res;
        }
        FPS& operator++() { return ++safe_get(0), * this; }
        FPS& operator--() { return --safe_get(0), * this; }
        FPS operator++(int) {
            FPS res = *this;
            ++(*this);
            return res;
        }
        FPS operator--(int) {
            FPS res = *this;
            --(*this);
            return res;
        }

        /* Binary Operations With Constant */

        FPS& operator+=(const value_type& x) { return safe_get(0) += x, *this; }
        FPS& operator-=(const value_type& x) { return safe_get(0) -= x, *this; }
        FPS& operator*=(const value_type& x) {
            for (auto& e : *this) e *= x;
            return *this;
        }
        FPS& operator/=(const value_type& x) { return *this *= x.inv(); }

        friend FPS operator+(FPS f, const value_type& x) { f += x; return f; }
        friend FPS operator+(const value_type& x, FPS f) { f += x; return f; }
        friend FPS operator-(FPS f, const value_type& x) { f -= x; return f; }
        friend FPS operator-(const value_type& x, FPS f) { f -= x; return -f; }
        friend FPS operator*(FPS f, const value_type& x) { f *= x; return f; }
        friend FPS operator*(const value_type& x, FPS f) { f *= x; return f; }
        friend FPS operator/(FPS f, const value_type& x) { f /= x; return f; }

        /* Binary Operations With Formal Power Series */

        FPS& operator+=(const FPS& g) {
            const int n = g.size();
            ensure(n);
            for (int i = 0; i < n; ++i) (*this)[i] += g[i];
            return *this;
        }
        FPS& operator-=(const FPS& g) {
            const int n = g.size();
            ensure(n);
            for (int i = 0; i < n; ++i) (*this)[i] -= g[i];
            return *this;
        }
        FPS& operator*=(const FPS& g) { return *this = *this * g; }
        FPS& operator/=(const FPS& g) { return *this = *this / g; }
        FPS& operator%=(const FPS& g) { return *this = *this % g; }

        friend FPS operator+(FPS f, const FPS& g) { f += g; return f; }
        friend FPS operator-(FPS f, const FPS& g) { f -= g; return f; }
        friend FPS operator*(const FPS& f, const FPS& g) { return mult(f, g); }
        friend FPS operator/(FPS f, FPS g) {
            if (f.size() < 60) return FPSNaive<mint>(f).div_mod(g).first;
            f.cut_trailing_zeros(), g.cut_trailing_zeros();
            const int fd = f.deg(), gd = g.deg();
            assert(gd >= 0);
            if (fd < gd) return {};
            if (gd == 0) {
                f /= g[0];
                return f;
            }
            std::reverse(f.begin(), f.end()), std::reverse(g.begin(), g.end());
            const int qd = fd - gd;
            FPS q = f * g.inv(qd + 1);
            q.cut(qd + 1);
            std::reverse(q.begin(), q.end());
            return q;
        }
        friend FPS operator%(const FPS& f, const FPS& g) { return f.div_mod(g).second; }
        std::pair<FPS, FPS> div_mod(const FPS& g) const {
            if (size() < 60) {
                auto [q, r] = FPSNaive<mint>(*this).div_mod(g);
                return { q, r };
            }
            FPS q = *this / g, r = *this - g * q;
            r.cut_trailing_zeros();
            return { q, r };
        }

        /* Shift Operations */

        FPS& operator<<=(const int shamt) {
            return this->insert(this->begin(), shamt, 0), * this;
        }
        FPS& operator>>=(const int shamt) {
            return this->erase(this->begin(), this->begin() + std::min(shamt, size())), * this;
        }
        friend FPS operator<<(FPS f, const int shamt) { f <<= shamt; return f; }
        friend FPS operator>>(FPS f, const int shamt) { f >>= shamt; return f; }

        /* Compare */

        friend bool operator==(const FPS& f, const FPS& g) {
            const int n = f.size(), m = g.size();
            if (n < m) return g == f;
            for (int i = 0; i < m; ++i) if (f[i] != g[i]) return false;
            for (int i = m; i < n; ++i) if (f[i] != 0) return false;
            return true;
        }
        friend bool operator!=(const FPS& f, const FPS& g) { return not (f == g); }

        /* Other Operations */

        FPS& diff_inplace() {
            const int n = size();
            for (int i = 1; i < n; ++i) (*this)[i - 1] = (*this)[i] * i;
            return (*this)[n - 1] = 0, *this;
        }
        FPS diff() const {
            FPS res = *this;
            res.diff_inplace();
            return res;
        }
        FPS& intg_inplace() {
            const int n = size();
            inv_mods<value_type> invs(n);
            this->resize(n + 1);
            for (int i = n; i > 0; --i) (*this)[i] = (*this)[i - 1] * invs[i];
            return (*this)[0] = 0, *this;
        }
        FPS intg() const {
            FPS res = *this;
            res.intg_inplace();
            return res;
        }
        
        FPS& inv_inplace(const int n = -1) { return *this = inv(n); } 
        FPS inv(int n = -1) const {
            if (n < 0) n = size();
            if (n < 60) return FPSNaive<mint>(*this).inv();
            if (auto sp_f = sparse_fps_format(15); sp_f.has_value()) return inv_sparse(std::move(*sp_f), n);
            FPS res{ (*this)[0].inv() };
            for (int k = 1; k < n; k *= 2) {
                FPS tmp(cut_copy(k * 2) * (res * res));
                tmp.resize(2 * k);
                res = 2 * res - tmp;
            }
            res.resize(n);
            return res;
        }
        FPS& log_inplace(int n = -1) { return *this = log(n); }
        FPS log(int n = -1) const {
            assert(safe_get(0) == 1);
            if (n < 0) n = size();
            if (n < 60) return FPSNaive<mint>(cut_copy(n)).log();
            if (auto sp_f = sparse_fps_format(15); sp_f.has_value()) return log_sparse(std::move(*sp_f), n);
            FPS res = inv(n) * diff();
            res.resize(n - 1);
            return res.intg();
        }
        FPS& exp_inplace(int n = -1) { return *this = exp(n); }
        FPS exp(int n = -1) {
            assert(safe_get(0) == 0);
            if (n < 0) n = size();
            if (n < 60) return FPSNaive<mint>(cut_copy(n)).exp();
            if (auto sp_f = sparse_fps_format(15); sp_f.has_value()) return exp_sparse(std::move(*sp_f), n);
            FPS res{ 1 };
            for (int k = 1; k < n; k *= 2) res *= ++(cut_copy(k * 2) - res.log(k * 2)), res.cut(k * 2);
            res.resize(n);
            return res;
        }
        FPS& pow_inplace(long long k, int n = -1) { return *this = pow(k, n); }
        FPS pow(const long long k, int n = -1) const {
            if (n < 0) n = size();
            if (n < 60) return FPSNaive<mint>(cut_copy(n)).pow(k);
            if (auto sp_f = sparse_fps_format(15); sp_f.has_value()) return pow_sparse(std::move(*sp_f), k, n);
            if (k == 0) {
                FPS f{ 1 };
                f.resize(n);
                return f;
            }
            int tlz = 0;
            while (tlz < size() and (*this)[tlz] == 0) ++tlz;
            if (tlz == size() or tlz > (n - 1) / k) return FPS(n, 0);
            const int m = n - tlz * k;
            FPS f = *this >> tlz;
            value_type base = f[0];
            return ((((f /= base).log(m) *= k).exp(m) *= base.pow(k)) <<= (tlz * k));
        }
        std::optional<FPS> safe_sqrt(int n = -1) const {
            if (n < 0) n = size();
            if (n < 60) return FPSNaive<mint>(cut_copy(n)).safe_sqrt();
            if (auto sp_f = sparse_fps_format(15); sp_f.has_value()) return safe_sqrt_sparse(std::move(*sp_f), n);
            int tlz = 0;
            while (tlz < size() and (*this)[tlz] == 0) ++tlz;
            if (tlz == size()) return FPS(n, 0);
            if (tlz & 1) return std::nullopt;
            const int m = n - tlz / 2;

            FPS h(this->begin() + tlz, this->end());
            auto q0 = ::safe_sqrt(h[0]);
            if (not q0.has_value()) return std::nullopt;
            FPS res{ *q0 };
            mint inv_2 = mint(2).inv();
            for (int k = 1; k < m; k *= 2) {
                FPS tmp = h.cut_copy(k * 2) * res.inv(2 * k);
                tmp.cut(2 * k);
                res += tmp, res *= inv_2;
            }
            res.resize(m);
            res <<= tlz / 2;
            return res;
        }
        FPS& sqrt_inplace(int n = -1) { return *this = sqrt(n); }
        FPS sqrt(int n = -1) const {
            return *safe_sqrt(n);
        }

        mint eval(mint x) const {
            mint y = 0;
            for (int i = size() - 1; i >= 0; --i) y = y * x + (*this)[i];
            return y;
        }

        static FPS prod(const std::vector<FPS>& fs) {
            auto comp = [](const FPS& f, const FPS& g) { return f.size() > g.size(); };
            std::priority_queue<FPS, std::vector<FPS>, decltype(comp)> pq{ comp };
            for (const auto& f : fs) pq.push(f);
            while (pq.size() > 1) {
                auto f = pq.top();
                pq.pop();
                auto g = pq.top();
                pq.pop();
                pq.push(f * g);
            }
            return pq.top();
        }

    protected:
        static convolution_t<mint> mult;

        std::optional<std::vector<std::pair<int, value_type>>> sparse_fps_format(int max_size) const {
            std::vector<std::pair<int, value_type>> res;
            for (int i = 0; i <= deg() and int(res.size()) <= max_size; ++i) if (value_type v = (*this)[i]; v != 0) res.emplace_back(i, v);
            if (int(res.size()) > max_size) return std::nullopt;
            return res;
        }

        static FPS div_fps_sparse(const FPS& f, const std::vector<std::pair<int, value_type>>& g, int n) {
            const int siz = g.size();
            assert(siz and g[0].first == 0);
            const value_type inv_g0 = g[0].second.inv();
            FPS h(n);
            for (int i = 0; i < n; ++i) {
                value_type v = f.safe_get(i);
                for (int idx = 1; idx < siz; ++idx) {
                    const auto& [j, gj] = g[idx];
                    if (j > i) break;
                    v -= gj * h[i - j];
                }
                h[i] = v * inv_g0;
            }
            return h;
        }
        static FPS inv_sparse(const std::vector<std::pair<int, value_type>>& g, const int n) {
            return div_fps_sparse(FPS{ 1 }, g, n);
        }
        static FPS exp_sparse(const std::vector<std::pair<int, value_type>>& f, const int n) {
            const int siz = f.size();
            assert(not siz or f[0].first != 0);
            FPS g(n);
            g[0] = 1;
            inv_mods<value_type> invs(n);
            for (int i = 1; i < n; ++i) {
                value_type v = 0;
                for (const auto& [j, fj] : f) {
                    if (j > i) break;
                    v += j * fj * g[i - j];
                }
                v *= invs[i];
                g[i] = v;
            }
            return g;
        }
        static FPS log_sparse(const std::vector<std::pair<int, value_type>>& f, const int n) {
            const int siz = f.size();
            assert(siz and f[0].first == 0 and f[0].second == 1);
            FPS g(n);
            for (int idx = 1; idx < siz; ++idx) {
                const auto& [j, fj] = f[idx];
                if (j >= n) break;
                g[j] = j * fj;
            }
            inv_mods<value_type> invs(n);
            for (int i = 1; i < n; ++i) {
                value_type v = g[i];
                for (int idx = 1; idx < siz; ++idx) {
                    const auto& [j, fj] = f[idx];
                    if (j > i) break;
                    v -= fj * g[i - j] * (i - j);
                }
                v *= invs[i];
                g[i] = v;
            }
            return g;
        }
        static FPS pow_sparse(const std::vector<std::pair<int, value_type>>& f, const long long k, const int n) {
            if (k == 0) {
                FPS res(n, 0);
                res[0] = 1;
                return res;
            }
            const int siz = f.size();
            if (not siz) return FPS(n, 0);
            const int p = f[0].first;
            if (p > (n - 1) / k) return FPS(n, 0);
            const value_type inv_f0 = f[0].second.inv();
            const int lz = p * k;
            FPS g(n);
            g[lz] = f[0].second.pow(k);
            inv_mods<value_type> invs(n);
            for (int i = 1; lz + i < n; ++i) {
                value_type v = 0;
                for (int idx = 1; idx < siz; ++idx) {
                    auto [j, fj] = f[idx];
                    j -= p;
                    if (j > i) break;
                    v += fj * g[lz + i - j] * (value_type(k) * j - (i - j));
                }
                v *= invs[i] * inv_f0;
                g[lz + i] = v;
            }
            return g;
        }
        static std::optional<FPS> safe_sqrt_sparse(const std::vector<std::pair<int, value_type>>& f, const int n) {
            const int siz = f.size();
            if (not siz) return FPS(n, 0);
            const int p = f[0].first;
            if (p % 2 == 1) return std::nullopt;
            if (p / 2 >= n) return FPS(n, 0);
            const value_type inv_f0 = f[0].second.inv();
            const int lz = p / 2;
            FPS g(n);
            auto opt_g0 = ::safe_sqrt(f[0].second);
            if (not opt_g0.has_value()) return std::nullopt;
            g[lz] = *opt_g0;
            value_type k = mint(2).inv();
            inv_mods<value_type> invs(n);
            for (int i = 1; lz + i < n; ++i) {
                value_type v = 0;
                for (int idx = 1; idx < siz; ++idx) {
                    auto [j, fj] = f[idx];
                    j -= p;
                    if (j > i) break;
                    v += fj * g[lz + i - j] * (k * j - (i - j));
                }
                v *= invs[i] * inv_f0;
                g[lz + i] = v;
            }
            return g;
        }
        static FPS sqrt_sparse(const std::vector<std::pair<int, value_type>>& f, const int n) {
            return *safe_sqrt(f, n);
        }
    };

    template <typename mint>
    convolution_t<mint> FPS<mint>::mult = [](const auto&, const auto&) {
        std::cerr << "convolution function is not available." << std::endl;
        assert(false);
        return std::vector<mint>{};
    };

} // namespace suisen

template <typename mint>
suisen::FPS<mint> sqrt(suisen::FPS<mint> a) {
    return a.sqrt();
}
template <typename mint>
suisen::FPS<mint> log(suisen::FPS<mint> a) {
    return a.log();
}
template <typename mint>
suisen::FPS<mint> exp(suisen::FPS<mint> a) {
    return a.exp();
}
template <typename mint, typename T>
suisen::FPS<mint> pow(suisen::FPS<mint> a, T b) {
    return a.pow(b);
}
template <typename mint>
suisen::FPS<mint> inv(suisen::FPS<mint> a) {
    return a.inv();
}

namespace suisen {
    template <typename T, typename U = T>
    struct factorial {
        factorial() {}
        factorial(int n) { ensure(n); }

        static void ensure(const int n) {
            int sz = _fac.size();
            if (n + 1 <= sz) return;
            int new_size = std::max(n + 1, sz * 2);
            _fac.resize(new_size), _fac_inv.resize(new_size);
            for (int i = sz; i < new_size; ++i) _fac[i] = _fac[i - 1] * i;
            _fac_inv[new_size - 1] = U(1) / _fac[new_size - 1];
            for (int i = new_size - 1; i > sz; --i) _fac_inv[i - 1] = _fac_inv[i] * i;
        }

        T fac(const int i) {
            ensure(i);
            return _fac[i];
        }
        T operator()(int i) {
            return fac(i);
        }
        U fac_inv(const int i) {
            ensure(i);
            return _fac_inv[i];
        }
        U binom(const int n, const int r) {
            if (n < 0 or r < 0 or n < r) return 0;
            ensure(n);
            return _fac[n] * _fac_inv[r] * _fac_inv[n - r];
        }
        U perm(const int n, const int r) {
            if (n < 0 or r < 0 or n < r) return 0;
            ensure(n);
            return _fac[n] * _fac_inv[n - r];
        }
    private:
        static std::vector<T> _fac;
        static std::vector<U> _fac_inv;
    };
    template <typename T, typename U>
    std::vector<T> factorial<T, U>::_fac{ 1 };
    template <typename T, typename U>
    std::vector<U> factorial<T, U>::_fac_inv{ 1 };
} // namespace suisen

#include <cstdint>

#include <numeric>

namespace suisen {
// referece: https://37zigen.com/linear-sieve/
class LinearSieve {
    public:
        LinearSieve(const int n) : _n(n), min_prime_factor(std::vector<int>(n + 1)) {
            std::iota(min_prime_factor.begin(), min_prime_factor.end(), 0);
            prime_list.reserve(_n / 20);
            for (int d = 2; d <= _n; ++d) {
                if (min_prime_factor[d] == d) prime_list.push_back(d);
                const int prime_max = std::min(min_prime_factor[d], _n / d);
                for (int prime : prime_list) {
                    if (prime > prime_max) break;
                    min_prime_factor[prime * d] = prime;
                }
            }
        }
        int prime_num() const noexcept { return prime_list.size(); }
        /**
         * Returns a vector of primes in [0, n].
         * It is guaranteed that the returned vector is sorted in ascending order.
         */
        const std::vector<int>& get_prime_list() const noexcept  {
            return prime_list;
        }
        const std::vector<int>& get_min_prime_factor() const noexcept { return min_prime_factor; }
        /**
         * Returns a vector of `{ prime, index }`.
         * It is guaranteed that the returned vector is sorted in ascending order.
         */
        std::vector<std::pair<int, int>> factorize(int n) const noexcept {
            assert(0 < n and n <= _n);
            std::vector<std::pair<int, int>> prime_powers;
            while (n > 1) {
                int p = min_prime_factor[n], c = 0;
                do { n /= p, ++c; } while (n % p == 0);
                prime_powers.emplace_back(p, c);
            }
            return prime_powers;
        }
    private:
        const int _n;
        std::vector<int> min_prime_factor;
        std::vector<int> prime_list;
};
} // namespace suisen

namespace suisen {
    // returns { 0^k, 1^k, ..., n^k }
    template <typename mint>
    std::vector<mint> powers(uint32_t n, uint64_t k) {
        const auto mpf = LinearSieve(n).get_min_prime_factor();
        std::vector<mint> res(n + 1);
        res[0] = k == 0;
        for (uint32_t i = 1; i <= n; ++i) res[i] = i == 1 ? 1 : uint32_t(mpf[i]) == i ? mint(i).pow(k) : res[mpf[i]] * res[i / mpf[i]];
        return res;
    }
} // namespace suisen

// reference: https://en.wikipedia.org/wiki/Eulerian_number
namespace suisen {
    template <typename FPSType>
    std::vector<typename FPSType::value_type> eulerian_number(uint32_t n) {
        using mint = typename FPSType::value_type;
        if (n == 0) return {};
        factorial<mint> fac(n + 1);
        const uint32_t h = (n + 1) >> 1;
        FPSType f = powers<mint>(h, n);
        f.erase(f.begin());
        FPSType g(h);
        for (uint32_t i = 0; i < h; ++i) {
            mint v = fac.binom(n + 1, i);
            g[i] = i & 1 ? -v : v;
        }
        FPSType res = f * g;
        res.resize(n);
        for (uint32_t i = h; i < n; ++i) res[i] = res[n - 1 - i];
        return res;
    }
    template <typename mint>
    std::vector<std::vector<mint>> eulerian_number_table(uint32_t n) {
        if (n == 0) return {};
        std::vector dp(n + 1, std::vector<mint>{});
        for (uint32_t i = 1; i <= n; ++i) {
            dp[i].resize(i);
            dp[i][0] = dp[i][i - 1] = 1;
            for (uint32_t j = 1; j < i - 1; ++j) dp[i][j] = (i - j) * dp[i - 1][j - 1] + (j + 1) * dp[i - 1][j];
        }
        return dp;
    }
} // namespace suisen

/**
 * [Idea] reference : https://motsu-xe.hatenablog.com/entry/2021/05/13/224016
 * 
 * SWAG + simulate a deque with 2 stacks
 * 
 * [Operations] reference : https://www.slideshare.net/catupper/amortize-analysis-of-deque-with-2-stack
 * 
 * `l`, `r` is a stack of { value, sum }
 * 
 *     accumulate
 *    <----------  ------> fold values from inside
 *   (     l     ][  r    )
 * 
 * pop_front:
 *  1. `l` is not empty
 *    (   l ][  r  )   ->   ( l ][  r  )      # pop from `l`. O(1)
 *  2. `l` is empty
 *    (][    r     )   -> (   l ][  r  )      # split `r` at its middle point. amortized O(1)
 *    (   l ][  r  )   ->   ( l ][  r  )      # pop from `l`. O(1)
 * 
 * pop_back:
 *  1. `r` is not empty
 *    (  l  ][ r   )   ->   (   l ][ r )      # pop from `r`. O(1)
 *  2. `r` is empty
 *    (     l    ][)   ->   (  l  ][ r   )    # split `l` at its middle point. amortized O(1)
 *    (  l  ][ r   )   ->   (  l  ][ r )      # pop from `r`. O(1)
 * 
 * push_front:
 *    (  l  ][  r  )   -> (    l  ][  r  )    # push to `l`. O(1)
 * 
 * push_back:
 *    (  l  ][  r  )   ->   (  l  ][  r    )  # push to `r`. O(1)
 */

namespace suisen {
    template <typename T, T(*op)(T, T), T(*e)()>
    struct DequeAggregation {
        struct DequeAggregationIterator {
            using difference_type = int;
            using value_type = T;
            using pointer = value_type*;
            using reference = value_type&;
            using iterator_category = std::random_access_iterator_tag;

            using fi_iterator_type = typename std::vector<std::pair<value_type, value_type>>::const_reverse_iterator;
            using se_iterator_type = typename std::vector<std::pair<value_type, value_type>>::const_iterator;

            fi_iterator_type it_l;
            fi_iterator_type it_l_end;
            se_iterator_type it_r_begin;
            se_iterator_type it_r;

            DequeAggregationIterator& operator++() {
                if (it_l == it_l_end) ++it_r;
                else ++it_l;
                return *this;
            }
            DequeAggregationIterator operator++(int) { DequeAggregationIterator ret = *this; ++(*this); return ret; }
            DequeAggregationIterator& operator--() {
                if (it_r == it_r_begin) --it_l;
                else --it_r;
                return *this;
            }
            DequeAggregationIterator operator--(int) { DequeAggregationIterator ret = *this; --(*this); return ret; }
            DequeAggregationIterator& operator+=(difference_type dif) {
                if (dif < 0) return *this -= -dif;
                if (int d = it_l_end - it_l; d < dif) it_l = it_l_end, it_r += dif - d;
                else it_l += dif;
                return *this;
            }
            friend DequeAggregationIterator operator+(DequeAggregationIterator it, difference_type dif) { it += dif; return it; }
            friend DequeAggregationIterator operator+(difference_type dif, DequeAggregationIterator it) { it += dif; return it; }
            DequeAggregationIterator& operator-=(difference_type dif) {
                if (dif < 0) return *this += -dif;
                if (int d = it_r - it_r_begin; d < dif) it_r = it_r_begin, it_l -= dif - d;
                else it_r -= dif;
                return *this;
            }
            friend DequeAggregationIterator operator-(DequeAggregationIterator it, difference_type dif) { it -= dif; return it; }
            difference_type operator-(const DequeAggregationIterator &rhs) const {
                difference_type d1 = it_l == it_l_end ? it_r - it_r_begin : it_l - it_l_end;
                difference_type d2 = rhs.it_l == rhs.it_l_end ? rhs.it_r - rhs.it_r_begin : rhs.it_l - rhs.it_l_end;
                return d1 - d2;
            }
            const value_type& operator[](difference_type i) const { return *((*this) + i); }
            const value_type& operator*() const { return it_l == it_l_end ? it_r->first : it_l->first; }
            bool operator!=(const DequeAggregationIterator &rhs) const { return it_l != rhs.it_l or it_r != rhs.it_r; }
            bool operator==(const DequeAggregationIterator &rhs) const { return not (*this != rhs); }
            bool operator< (const DequeAggregationIterator &rhs) const { return (*this) - rhs <  0; }
            bool operator<=(const DequeAggregationIterator &rhs) const { return (*this) - rhs <= 0; }
            bool operator> (const DequeAggregationIterator &rhs) const { return (*this) - rhs >  0; }
            bool operator>=(const DequeAggregationIterator &rhs) const { return (*this) - rhs >= 0; }
        };
        
        using iterator = DequeAggregationIterator;
        using difference_type = typename iterator::difference_type;
        using value_type = typename iterator::value_type;
        using pointer = typename iterator::pointer;
        using reference = typename iterator::reference;

        DequeAggregation() = default;
        template <typename InputIterator, std::enable_if_t<std::is_constructible_v<value_type, typename InputIterator::value_type>, std::nullptr_t> = nullptr>
        DequeAggregation(InputIterator first, InputIterator last) {
            for (; first != last; ++first) push_back(*first);
        }
        template <typename Container, std::enable_if_t<std::is_constructible_v<value_type, typename Container::value_type>, std::nullptr_t> = nullptr>
        DequeAggregation(const Container &c) : DequeAggregation(std::begin(c), std::end(c)) {}

        value_type prod() const {
            return op(prod(_st_l), prod(_st_r));
        }
        void push_back(const value_type &val) { _st_r.emplace_back(val, op(prod(_st_r), val)); }
        void push_front(const value_type &val) { _st_l.emplace_back(val, op(val, prod(_st_l))); }
        void pop_back() {
            if (_st_r.size()) return _st_r.pop_back();
            const int siz = _st_l.size();
            const int l = siz >> 1, r = siz - l;
            assert(r); // <=> siz > 0
            for (int i = r - 1; i > 0; --i) push_back(std::move(_st_l[i].first));
            _st_l.erase(_st_l.begin(), _st_l.begin() + r);
            if (l == 0) return;
            _st_l[0].second = _st_l[0].first;
            for (int i = 1; i < l; ++i) _st_l[i].second = op(_st_l[i].first, _st_l[i - 1].second);
        }
        void pop_front() {
            if (_st_l.size()) return _st_l.pop_back();
            const int siz = _st_r.size();
            const int r = siz >> 1, l = siz - r;
            assert(l); // <=> siz > 0
            for (int i = l - 1; i > 0; --i) push_front(std::move(_st_r[i].first));
            _st_r.erase(_st_r.begin(), _st_r.begin() + l);
            if (r == 0) return;
            _st_r[0].second = _st_r[0].first;
            for (int i = 1; i < r; ++i) _st_r[i].second = op(_st_r[i - 1].second, _st_r[i].first);
        }
        const value_type& front() const { return _st_l.size() ? _st_l.back().first : _st_r.front().first; }
        const value_type& back() const { return _st_r.size() ? _st_r.back().first : _st_l.front().first; }
        const value_type& operator[](int i) const {
            const int k = i - _st_l.size();
            return k < 0 ? _st_l[~k].first : _st_r[k].first;
        }

        int size() const { return _st_l.size() + _st_r.size(); }
        void clear() { _st_l.clear(), _st_r.clear(); }
        void shrink_to_fit() { _st_l.shrink_to_fit(), _st_r.shrink_to_fit(); }

        iterator begin() const { return iterator { _st_l.rbegin(), _st_l.rend(), _st_r.begin(), _st_r.begin() }; }
        iterator end() const { return iterator { _st_l.rend(), _st_l.rend(), _st_r.begin(), _st_r.end() }; }
        iterator cbegin() const { return begin(); }
        iterator cend() const { return end(); }

    private:
        std::vector<std::pair<value_type, value_type>> _st_l, _st_r;

        value_type prod(const std::vector<std::pair<value_type, value_type>> &st) const {
            return st.empty() ? e() : st.back().second;
        }
    };
} // namespace suisen

mint op(mint x, mint y) {
    return x * y;
}
mint e() {
    return 1;
}

constexpr uint32_t K_MAX = 5000;

int main() {
    suisen::FPS<mint>::set_multiplication([](const auto &a, const auto &b) { return atcoder::convolution(a, b); });

    std::ios::sync_with_stdio(false);
    std::cin.tie(nullptr);

    uint32_t n;
    uint64_t m;
    std::cin >> n >> m;

    std::vector<mint> c(K_MAX + 1);
    for (uint32_t i = 0; i < n; ++i) {
        uint32_t k;
        std::cin >> k;
        ++c[k];
    }
    suisen::factorial<mint> fac(n + K_MAX);

    mint ans = 0;

    suisen::DequeAggregation<mint, op, e> dq;
    for (uint32_t d = 0; d < n; ++d) dq.push_front(m + d);
    for (uint32_t k = 1; k <= K_MAX; ++k) {
        std::vector<mint> e = suisen::eulerian_number<suisen::FPS<mint>>(k);
        dq.push_front(m + n + k - 1);
        mint sum = 0;
        const uint32_t p = std::min(uint64_t(k), m);
        for (uint32_t i = 0; i < p; ++i) {
            sum += e[i] * dq.prod();
            dq.pop_front();
            dq.push_back(m - i - 1);
        }
        ans += c[k] * sum * fac.fac_inv(n + k);
        for (uint32_t i = p; i --> 0;) {
            dq.push_front(m - i + n + k - 1);
            dq.pop_back();
        }
    }
    std::cout << ans.val() << std::endl;

    return 0;
}