#include <bits/stdc++.h>
using namespace std;
typedef long long ll;
template<class T>bool chmax(T &a, const T &b) { if (a<b) { a=b; return true; } return false; }
template<class T>bool chmin(T &a, const T &b) { if (b<a) { a=b; return true; } return false; }
#define all(x) (x).begin(),(x).end()
#define fi first
#define se second
#define mp make_pair
#define si(x) int(x.size())
const int mod=998244353,MAX=300005,INF=1<<30;

// FPS 全部載せ

// from: https://gist.github.com/yosupo06/ddd51afb727600fd95d9d8ad6c3c80c9
// (based on AtCoder STL)

#include <algorithm>
#include <array>

#ifdef _MSC_VER
#include <intrin.h>
#endif

namespace atcoder {
    
    namespace internal {
        
        int ceil_pow2(int n) {
            int x = 0;
            while ((1U << x) < (unsigned int)(n)) x++;
            return x;
        }
        
        int bsf(unsigned int n) {
#ifdef _MSC_VER
            unsigned long index;
            _BitScanForward(&index, n);
            return index;
#else
            return __builtin_ctz(n);
#endif
        }
        
    }  // namespace internal
    
}  // namespace atcoder



#include <utility>

namespace atcoder {
    
    namespace internal {
        
        constexpr long long safe_mod(long long x, long long m) {
            x %= m;
            if (x < 0) x += m;
            return x;
        }
        
        struct barrett {
            unsigned int _m;
            unsigned long long im;
            
            barrett(unsigned int m) : _m(m), im((unsigned long long)(-1) / m + 1) {}
            
            unsigned int umod() const { return _m; }
            
            unsigned int mul(unsigned int a, unsigned int b) const {
                
                unsigned long long z = a;
                z *= b;
#ifdef _MSC_VER
                unsigned long long x;
                _umul128(z, im, &x);
#else
                unsigned long long x =
                (unsigned long long)(((unsigned __int128)(z)*im) >> 64);
#endif
                unsigned int v = (unsigned int)(z - x * _m);
                if (_m <= v) v += _m;
                return v;
            }
        };
        
        constexpr long long pow_mod_constexpr(long long x, long long n, int m) {
            if (m == 1) return 0;
            unsigned int _m = (unsigned int)(m);
            unsigned long long r = 1;
            unsigned long long y = safe_mod(x, m);
            while (n) {
                if (n & 1) r = (r * y) % _m;
                y = (y * y) % _m;
                n >>= 1;
            }
            return r;
        }
        
        constexpr bool is_prime_constexpr(int n) {
            if (n <= 1) return false;
            if (n == 2 || n == 7 || n == 61) return true;
            if (n % 2 == 0) return false;
            long long d = n - 1;
            while (d % 2 == 0) d /= 2;
            for (long long a : {2, 7, 61}) {
                long long t = d;
                long long y = pow_mod_constexpr(a, t, n);
                while (t != n - 1 && y != 1 && y != n - 1) {
                    y = y * y % n;
                    t <<= 1;
                }
                if (y != n - 1 && t % 2 == 0) {
                    return false;
                }
            }
            return true;
        }
        template <int n> constexpr bool is_prime = is_prime_constexpr(n);
        
        constexpr std::pair<long long, long long> inv_gcd(long long a, long long b) {
            a = safe_mod(a, b);
            if (a == 0) return {b, 0};
            
            long long s = b, t = a;
            long long m0 = 0, m1 = 1;
            
            while (t) {
                long long u = s / t;
                s -= t * u;
                m0 -= m1 * u;  // |m1 * u| <= |m1| * s <= b
                
                
                auto tmp = s;
                s = t;
                t = tmp;
                tmp = m0;
                m0 = m1;
                m1 = tmp;
            }
            if (m0 < 0) m0 += b / s;
            return {s, m0};
        }
        
        constexpr int primitive_root_constexpr(int m) {
            if (m == 2) return 1;
            if (m == 167772161) return 3;
            if (m == 469762049) return 3;
            if (m == 754974721) return 11;
            if (m == 998244353) return 3;
            int divs[20] = {};
            divs[0] = 2;
            int cnt = 1;
            int x = (m - 1) / 2;
            while (x % 2 == 0) x /= 2;
            for (int i = 3; (long long)(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_constexpr(g, (m - 1) / divs[i], m) == 1) {
                        ok = false;
                        break;
                    }
                }
                if (ok) return g;
            }
        }
        template <int m> constexpr int primitive_root = primitive_root_constexpr(m);
        
    }  // namespace internal
    
}  // namespace atcoder


#include <cassert>
#include <numeric>
#include <type_traits>

namespace atcoder {
    
    namespace internal {
        
#ifndef _MSC_VER
        template <class T>
        using is_signed_int128 =
        typename std::conditional<std::is_same<T, __int128_t>::value ||
        std::is_same<T, __int128>::value,
        std::true_type,
        std::false_type>::type;
        
        template <class T>
        using is_unsigned_int128 =
        typename std::conditional<std::is_same<T, __uint128_t>::value ||
        std::is_same<T, unsigned __int128>::value,
        std::true_type,
        std::false_type>::type;
        
        template <class T>
        using make_unsigned_int128 =
        typename std::conditional<std::is_same<T, __int128_t>::value,
        __uint128_t,
        unsigned __int128>;
        
        template <class T>
        using is_integral = typename std::conditional<std::is_integral<T>::value ||
        is_signed_int128<T>::value ||
        is_unsigned_int128<T>::value,
        std::true_type,
        std::false_type>::type;
        
        template <class T>
        using is_signed_int = typename std::conditional<(is_integral<T>::value &&
                                                         std::is_signed<T>::value) ||
        is_signed_int128<T>::value,
        std::true_type,
        std::false_type>::type;
        
        template <class T>
        using is_unsigned_int =
        typename std::conditional<(is_integral<T>::value &&
                                   std::is_unsigned<T>::value) ||
        is_unsigned_int128<T>::value,
        std::true_type,
        std::false_type>::type;
        
        template <class T>
        using to_unsigned = typename std::conditional<
        is_signed_int128<T>::value,
        make_unsigned_int128<T>,
        typename std::conditional<std::is_signed<T>::value,
        std::make_unsigned<T>,
        std::common_type<T>>::type>::type;
        
#else
        
        template <class T> using is_integral = typename std::is_integral<T>;
        
        template <class T>
        using is_signed_int =
        typename std::conditional<is_integral<T>::value && std::is_signed<T>::value,
        std::true_type,
        std::false_type>::type;
        
        template <class T>
        using is_unsigned_int =
        typename std::conditional<is_integral<T>::value &&
        std::is_unsigned<T>::value,
        std::true_type,
        std::false_type>::type;
        
        template <class T>
        using to_unsigned = typename std::conditional<is_signed_int<T>::value,
        std::make_unsigned<T>,
        std::common_type<T>>::type;
        
#endif
        
        template <class T>
        using is_signed_int_t = std::enable_if_t<is_signed_int<T>::value>;
        
        template <class T>
        using is_unsigned_int_t = std::enable_if_t<is_unsigned_int<T>::value>;
        
        template <class T> using to_unsigned_t = typename to_unsigned<T>::type;
        
    }  // namespace internal
    
}  // namespace atcoder

#include <cassert>
#include <numeric>
#include <type_traits>

#ifdef _MSC_VER
#include <intrin.h>
#endif

namespace atcoder {
    
    namespace internal {
        
        struct modint_base {};
        struct static_modint_base : modint_base {};
        
        template <class T> using is_modint = std::is_base_of<modint_base, T>;
        template <class T> using is_modint_t = std::enable_if_t<is_modint<T>::value>;
        
    }  // namespace internal
    
    template <int m, std::enable_if_t<(1 <= m)>* = nullptr>
    struct static_modint : internal::static_modint_base {
        using mint = static_modint;
        
    public:
        static constexpr int mod() { return m; }
        static mint raw(int v) {
            mint x;
            x._v = v;
            return x;
        }
        
        static_modint() : _v(0) {}
        template <class T, internal::is_signed_int_t<T>* = nullptr>
        static_modint(T v) {
            long long x = (long long)(v % (long long)(umod()));
            if (x < 0) x += umod();
            _v = (unsigned int)(x);
        }
        template <class T, internal::is_unsigned_int_t<T>* = nullptr>
        static_modint(T v) {
            _v = (unsigned int)(v % umod());
        }
        static_modint(bool v) { _v = ((unsigned int)(v) % umod()); }
        
        unsigned int val() const { return _v; }
        
        mint& operator++() {
            _v++;
            if (_v == umod()) _v = 0;
            return *this;
        }
        mint& operator--() {
            if (_v == 0) _v = umod();
            _v--;
            return *this;
        }
        mint operator++(int) {
            mint result = *this;
            ++*this;
            return result;
        }
        mint operator--(int) {
            mint result = *this;
            --*this;
            return result;
        }
        
        mint& operator+=(const mint& rhs) {
            _v += rhs._v;
            if (_v >= umod()) _v -= umod();
            return *this;
        }
        mint& operator-=(const mint& rhs) {
            _v -= rhs._v;
            if (_v >= umod()) _v += umod();
            return *this;
        }
        mint& operator*=(const mint& rhs) {
            unsigned long long z = _v;
            z *= rhs._v;
            _v = (unsigned int)(z % umod());
            return *this;
        }
        mint& operator/=(const mint& rhs) { return *this = *this * rhs.inv(); }
        
        mint operator+() const { return *this; }
        mint operator-() const { return mint() - *this; }
        
        mint pow(long long n) const {
            assert(0 <= n);
            mint x = *this, r = 1;
            while (n) {
                if (n & 1) r *= x;
                x *= x;
                n >>= 1;
            }
            return r;
        }
        mint inv() const {
            if (prime) {
                assert(_v);
                return pow(umod() - 2);
            } else {
                auto eg = internal::inv_gcd(_v, m);
                assert(eg.first == 1);
                return eg.second;
            }
        }
        
        friend mint operator+(const mint& lhs, const mint& rhs) {
            return mint(lhs) += rhs;
        }
        friend mint operator-(const mint& lhs, const mint& rhs) {
            return mint(lhs) -= rhs;
        }
        friend mint operator*(const mint& lhs, const mint& rhs) {
            return mint(lhs) *= rhs;
        }
        friend mint operator/(const mint& lhs, const mint& rhs) {
            return mint(lhs) /= rhs;
        }
        friend bool operator==(const mint& lhs, const mint& rhs) {
            return lhs._v == rhs._v;
        }
        friend bool operator!=(const mint& lhs, const mint& rhs) {
            return lhs._v != rhs._v;
        }
        
    private:
        unsigned int _v;
        static constexpr unsigned int umod() { return m; }
        static constexpr bool prime = internal::is_prime<m>;
    };
    
    template <int id> struct dynamic_modint : internal::modint_base {
        using mint = dynamic_modint;
        
    public:
        static int mod() { return (int)(bt.umod()); }
        static void set_mod(int m) {
            assert(1 <= m);
            bt = internal::barrett(m);
        }
        static mint raw(int v) {
            mint x;
            x._v = v;
            return x;
        }
        
        dynamic_modint() : _v(0) {}
        template <class T, internal::is_signed_int_t<T>* = nullptr>
        dynamic_modint(T v) {
            long long x = (long long)(v % (long long)(mod()));
            if (x < 0) x += mod();
            _v = (unsigned int)(x);
        }
        template <class T, internal::is_unsigned_int_t<T>* = nullptr>
        dynamic_modint(T v) {
            _v = (unsigned int)(v % mod());
        }
        dynamic_modint(bool v) { _v = ((unsigned int)(v) % mod()); }
        
        unsigned int val() const { return _v; }
        
        mint& operator++() {
            _v++;
            if (_v == umod()) _v = 0;
            return *this;
        }
        mint& operator--() {
            if (_v == 0) _v = umod();
            _v--;
            return *this;
        }
        mint operator++(int) {
            mint result = *this;
            ++*this;
            return result;
        }
        mint operator--(int) {
            mint result = *this;
            --*this;
            return result;
        }
        
        mint& operator+=(const mint& rhs) {
            _v += rhs._v;
            if (_v >= umod()) _v -= umod();
            return *this;
        }
        mint& operator-=(const mint& rhs) {
            _v += mod() - rhs._v;
            if (_v >= umod()) _v -= umod();
            return *this;
        }
        mint& operator*=(const mint& rhs) {
            _v = bt.mul(_v, rhs._v);
            return *this;
        }
        mint& operator/=(const mint& rhs) { return *this = *this * rhs.inv(); }
        
        mint operator+() const { return *this; }
        mint operator-() const { return mint() - *this; }
        
        mint pow(long long n) const {
            assert(0 <= n);
            mint x = *this, r = 1;
            while (n) {
                if (n & 1) r *= x;
                x *= x;
                n >>= 1;
            }
            return r;
        }
        mint inv() const {
            auto eg = internal::inv_gcd(_v, mod());
            assert(eg.first == 1);
            return eg.second;
        }
        
        friend mint operator+(const mint& lhs, const mint& rhs) {
            return mint(lhs) += rhs;
        }
        friend mint operator-(const mint& lhs, const mint& rhs) {
            return mint(lhs) -= rhs;
        }
        friend mint operator*(const mint& lhs, const mint& rhs) {
            return mint(lhs) *= rhs;
        }
        friend mint operator/(const mint& lhs, const mint& rhs) {
            return mint(lhs) /= rhs;
        }
        friend bool operator==(const mint& lhs, const mint& rhs) {
            return lhs._v == rhs._v;
        }
        friend bool operator!=(const mint& lhs, const mint& rhs) {
            return lhs._v != rhs._v;
        }
        
    private:
        unsigned int _v;
        static internal::barrett bt;
        static unsigned int umod() { return bt.umod(); }
    };
    template <int id> internal::barrett dynamic_modint<id>::bt = 998244353;
    
    using modint998244353 = static_modint<998244353>;
    using modint1000000007 = static_modint<1000000007>;
    using modint = dynamic_modint<-1>;
    
    namespace internal {
        
        template <class T>
        using is_static_modint = std::is_base_of<internal::static_modint_base, T>;
        
        template <class T>
        using is_static_modint_t = std::enable_if_t<is_static_modint<T>::value>;
        
        template <class> struct is_dynamic_modint : public std::false_type {};
        template <int id>
        struct is_dynamic_modint<dynamic_modint<id>> : public std::true_type {};
        
        template <class T>
        using is_dynamic_modint_t = std::enable_if_t<is_dynamic_modint<T>::value>;
        
    }  // namespace internal
    
}  // namespace atcoder

#include <cassert>
#include <type_traits>
#include <vector>

namespace atcoder {
    
    namespace internal {
        
        template <class mint, internal::is_static_modint_t<mint>* = nullptr>
        void butterfly(std::vector<mint>& a) {
            static constexpr int g = internal::primitive_root<mint::mod()>;
            int n = int(a.size());
            int h = internal::ceil_pow2(n);
            
            static bool first = true;
            static mint sum_e[30];  // sum_e[i] = ies[0] * ... * ies[i - 1] * es[i]
            if (first) {
                first = false;
                mint es[30], ies[30];  // es[i]^(2^(2+i)) == 1
                int cnt2 = bsf(mint::mod() - 1);
                mint e = mint(g).pow((mint::mod() - 1) >> cnt2), ie = e.inv();
                for (int i = cnt2; i >= 2; i--) {
                    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];
                }
            }
            for (int ph = 1; ph <= h; ph++) {
                int w = 1 << (ph - 1), p = 1 << (h - ph);
                mint now = 1;
                for (int s = 0; s < w; s++) {
                    int offset = s << (h - ph + 1);
                    for (int i = 0; i < p; i++) {
                        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[bsf(~(unsigned int)(s))];
                }
            }
        }
        
        template <class mint, internal::is_static_modint_t<mint>* = nullptr>
        void butterfly_inv(std::vector<mint>& a) {
            static constexpr int g = internal::primitive_root<mint::mod()>;
            int n = int(a.size());
            int h = internal::ceil_pow2(n);
            
            static bool first = true;
            static mint sum_ie[30];  // sum_ie[i] = es[0] * ... * es[i - 1] * ies[i]
            if (first) {
                first = false;
                mint es[30], ies[30];  // es[i]^(2^(2+i)) == 1
                int cnt2 = bsf(mint::mod() - 1);
                mint e = mint(g).pow((mint::mod() - 1) >> cnt2), ie = e.inv();
                for (int i = cnt2; i >= 2; i--) {
                    es[i - 2] = e;
                    ies[i - 2] = ie;
                    e *= e;
                    ie *= ie;
                }
                mint now = 1;
                for (int i = 0; i < cnt2 - 2; i++) {
                    sum_ie[i] = ies[i] * now;
                    now *= es[i];
                }
            }
            
            for (int ph = h; ph >= 1; ph--) {
                int w = 1 << (ph - 1), p = 1 << (h - ph);
                mint inow = 1;
                for (int s = 0; s < w; s++) {
                    int offset = s << (h - ph + 1);
                    for (int i = 0; i < p; i++) {
                        auto l = a[i + offset];
                        auto r = a[i + offset + p];
                        a[i + offset] = l + r;
                        a[i + offset + p] =
                        (unsigned long long)(mint::mod() + l.val() - r.val()) *
                        inow.val();
                    }
                    inow *= sum_ie[bsf(~(unsigned int)(s))];
                }
            }
        }
        
    }  // namespace internal
    
    template <class mint, internal::is_static_modint_t<mint>* = nullptr>
    std::vector<mint> convolution(std::vector<mint> a, std::vector<mint> b) {
        int n = int(a.size()), m = int(b.size());
        if (!n || !m) return {};
        if (std::min(n, m) <= 60) {
            if (n < m) {
                std::swap(n, m);
                std::swap(a, b);
            }
            std::vector<mint> ans(n + m - 1);
            for (int i = 0; i < n; i++) {
                for (int j = 0; j < m; j++) {
                    ans[i + j] += a[i] * b[j];
                }
            }
            return ans;
        }
        int z = 1 << internal::ceil_pow2(n + m - 1);
        a.resize(z);
        internal::butterfly(a);
        b.resize(z);
        internal::butterfly(b);
        for (int i = 0; i < z; i++) {
            a[i] *= b[i];
        }
        internal::butterfly_inv(a);
        a.resize(n + m - 1);
        mint iz = mint(z).inv();
        for (int i = 0; i < n + m - 1; i++) a[i] *= iz;
        return a;
    }
    
    template <unsigned int mod = 998244353,
    class T,
    std::enable_if_t<internal::is_integral<T>::value>* = nullptr>
    std::vector<T> convolution(const std::vector<T>& a, const std::vector<T>& b) {
        int n = int(a.size()), m = int(b.size());
        if (!n || !m) return {};
        
        using mint = static_modint<mod>;
        std::vector<mint> a2(n), b2(m);
        for (int i = 0; i < n; i++) {
            a2[i] = mint(a[i]);
        }
        for (int i = 0; i < m; i++) {
            b2[i] = mint(b[i]);
        }
        auto c2 = convolution(move(a2), move(b2));
        std::vector<T> c(n + m - 1);
        for (int i = 0; i < n + m - 1; i++) {
            c[i] = c2[i].val();
        }
        return c;
    }
    
    std::vector<long long> convolution_ll(const std::vector<long long>& a,
                                          const std::vector<long long>& b) {
        int n = int(a.size()), m = int(b.size());
        if (!n || !m) return {};
        
        static constexpr unsigned long long MOD1 = 754974721;  // 2^24
        static constexpr unsigned long long MOD2 = 167772161;  // 2^25
        static constexpr unsigned long long MOD3 = 469762049;  // 2^26
        static constexpr unsigned long long M2M3 = MOD2 * MOD3;
        static constexpr unsigned long long M1M3 = MOD1 * MOD3;
        static constexpr unsigned long long M1M2 = MOD1 * MOD2;
        static constexpr unsigned long long M1M2M3 = MOD1 * MOD2 * MOD3;
        
        static constexpr unsigned long long i1 =
        internal::inv_gcd(MOD2 * MOD3, MOD1).second;
        static constexpr unsigned long long i2 =
        internal::inv_gcd(MOD1 * MOD3, MOD2).second;
        static constexpr unsigned long long i3 =
        internal::inv_gcd(MOD1 * MOD2, MOD3).second;
        
        auto c1 = convolution<MOD1>(a, b);
        auto c2 = convolution<MOD2>(a, b);
        auto c3 = convolution<MOD3>(a, b);
        
        std::vector<long long> c(n + m - 1);
        for (int i = 0; i < n + m - 1; i++) {
            unsigned long long x = 0;
            x += (c1[i] * i1) % MOD1 * M2M3;
            x += (c2[i] * i2) % MOD2 * M1M3;
            x += (c3[i] * i3) % MOD3 * M1M2;
            long long diff =
            c1[i] - internal::safe_mod((long long)(x), (long long)(MOD1));
            if (diff < 0) diff += MOD1;
            static constexpr unsigned long long offset[5] = {
                0, 0, M1M2M3, 2 * M1M2M3, 3 * M1M2M3};
            x -= offset[diff % 5];
            c[i] = x;
        }
        
        return c;
    }
    
}  // namespace atcoder

using mint=atcoder::modint998244353;

vector<mint> prebat(vector<mint> S,int szsum){
    int z = 1 << atcoder::internal::ceil_pow2(szsum-1);
    auto res=S;
    res.resize(z);
    atcoder::internal::butterfly(res);
    return res;
}
// szsum = aの配列の長さ + bの配列の長さ

vector<mint> sufbat(vector<mint> S,int szsum){
    int z = 1 << atcoder::internal::ceil_pow2(szsum-1);
    auto res=S;
    atcoder::internal::butterfly_inv(res);
    res.resize(szsum-1);
    mint iz = mint(z).inv();
    for (int i = 0; i < szsum - 1; i++) res[i] *= iz;
    return res;
}
// szsum = aの配列の長さ + bの配列の長さ

mint inv[MAX],fac[MAX],finv[MAX];

void make(){
    
    fac[0]=fac[1]=1;
    finv[0]=finv[1]=1;
    inv[1]=1;
    
    for(int i=2;i<MAX;i++){
        inv[i]=-inv[mod%i]*(mod/i);
        fac[i]=fac[i-1]*i;
        finv[i]=finv[i-1]*inv[i];
    }
}

mint comb(ll a,ll b){
    if(a<b) return 0;
    return fac[a]*finv[b]*finv[a-b];
}

mint perm(ll a,ll b){
    if(a<b) return 0;
    return fac[a]*finv[a-b];
}

vector<mint> bibun(vector<mint> F,int deg){
    vector<mint> res(deg+1);
    for(int i=1;i<si(F)&&i-1<=deg;i++){
        res[i-1]=F[i]*i;
    }
    
    return res;
}

vector<mint> sekibun(vector<mint> F,int deg){
    vector<mint> res(deg+1);
    for(int i=0;i<min(si(F),deg);i++){
        res[i+1]=F[i]*inv[i+1];
    }
    
    return res;
}

vector<mint> invv(vector<mint> F,int deg){
    assert(F[0]!=0);
    
    mint kake=mint(F[0]).inv();
    for(int i=0;i<si(F);i++){
        F[i]*=kake;
    }
    vector<mint> G(1,1);
    int len=1;
    while(len<=deg){
        vector<mint> f=F;f.resize(len*2);
        vector<mint> g=G;g.resize(len*2);
        
        atcoder::internal::butterfly(f);
        atcoder::internal::butterfly(g);
        
        for(int i=0;i<len*2;i++) f[i]*=g[i];
        
        atcoder::internal::butterfly_inv(f);
        vector<mint> nf(len*2);
        for(int i=len;i<2*len;i++) nf[i-len]=f[i];
        
        f=nf;
        atcoder::internal::butterfly(f);
        
        for(int i=0;i<len*2;i++) f[i]*=g[i];
        
        atcoder::internal::butterfly_inv(f);
        
        mint iz=mint(len*2).inv();
        mint coe=-iz*iz;
        
        G.resize(len*2);
        
        for(int i=0;i<len;i++) G[len+i]=f[i]*coe;
        
        len*=2;
    }
    
    G.resize(deg+1);
    for(int i=0;i<=deg;i++) G[i]*=kake;
    
    return G;
}//1/Tのdeg次以下を返す

vector<mint> logg(vector<mint> F,int deg){
    assert(F[0]==1);
    
    vector<mint> FF=bibun(F,deg);
    vector<mint> waru=invv(F,deg);
    
    vector<mint> G=atcoder::convolution(FF,waru);
    
    G=sekibun(G,deg);
    
    return G;
}
// F0 = 1

vector<mint> expp(vector<mint> F,int deg){
    if(si(F)){
        assert(F[0]==0);
    }
    
    vector<mint> G(1,1);
    int len=1;
    while(len<=deg){
        vector<mint> nex=logg(G,len*2-1);
        for(int i=0;i<si(nex);i++) nex[i]*=(-1);
        for(int i=0;i<si(nex);i++){
            if(i<si(F)) nex[i]+=F[i];
        }
        nex[0]++;
        nex=atcoder::convolution(nex,G);
        nex.resize(len*2);
        
        len*=2;
        G=nex;
    }
    
    G.resize(deg+1);
    
    return G;
}
// F0 = 0

vector<mint> poww(vector<mint> F,int deg,ll K){
    if(K==0){
        vector<mint> res(deg+1);
        res[0]=1;
        return res;
    }
    if(si(F)==0){
        vector<mint> res(deg+1);
        return res;
    }
    
    ll geta=-1;
    mint kake=0;
    for(int i=0;i<si(F);i++){
        if(F[i]!=0){
            geta=i;
            kake=F[i].inv();
            break;
        }
    }
    
    if(geta==-1){
        vector<mint> res(deg+1);
        return res;
    }
    
    if(geta>1000000000LL/K){
        vector<mint> res(deg+1);
        return res;
    }
    if(geta*K>deg){
        vector<mint> res(deg+1);
        return res;
    }
    
    vector<mint> nF(si(F)-geta);
    for(int i=geta;i<si(F);i++){
        nF[i-geta]=(F[i]*kake);
    }
    
    F=nF;
    
    vector<mint> FF=logg(nF,deg-geta*K);
    for(int i=0;i<si(FF);i++) FF[i]*=K;
    
    vector<mint> G=expp(FF,deg-geta*K);
    
    kake=kake.inv();
    kake=kake.pow(K);
    
    vector<mint> res(deg+1);
    for(int i=0;i<si(G);i++){
        res[geta*K+i]=G[i]*kake;
    }
    
    return res;
}

mint senkeizenka(vector<mint> A,vector<mint> C,ll K){
    if(K<si(A)) return A[K];
    
    int D=si(A);
    assert(si(A)==si(C));
    vector<mint> Q(D+1);
    Q[0]=1;
    for(int i=1;i<=D;i++) Q[i]=-C[i-1];
    
    auto P=atcoder::convolution(A,Q);
    P.resize(D);
    
    while(K){
        auto Qneg=Q;
        for(int i=1;i<si(Qneg);i+=2) Qneg[i]=-Qneg[i];
        auto x=atcoder::convolution(P,Qneg);
        auto y=atcoder::convolution(Q,Qneg);
        
        P.clear();
        Q.clear();
        for(int i=(K&1);i<si(x);i+=2) P.push_back(x[i]);
        for(int i=0;i<si(y);i+=2) Q.push_back(y[i]);
        K/=2;
    }
    
    return P[0]/Q[0];
}
//a[0],...,a[d-1]
//c[1],...,c[d]

mint senkeizenka2(vector<mint> P,vector<mint> Q,ll K){
    
    while(K){
        auto Qneg=Q;
        for(int i=1;i<si(Qneg);i+=2) Qneg[i]=-Qneg[i];
        auto x=atcoder::convolution(P,Qneg);
        auto y=atcoder::convolution(Q,Qneg);
        
        P.clear();
        Q.clear();
        for(int i=(K&1);i<si(x);i+=2) P.push_back(x[i]);
        for(int i=0;i<si(y);i+=2) Q.push_back(y[i]);
        K/=2;
    }
    
    return P[0]/Q[0];
}
// P/Q

// make() を呼ばないとsekibun呼ぶやつで一部バグる
// MAX=2*deg ぐらい必要な気がする

pair<vector<mint>,vector<mint>> warizan(vector<mint> P,vector<mint> Q){
    if(si(P)<si(Q)) return mp(vector<mint>{},P);
    
    auto revP=P;reverse(all(revP));
    auto revQ=Q;reverse(all(revQ));
    revQ=invv(revQ,si(P)-si(Q));
    auto shou=atcoder::convolution(revP,revQ);
    shou.resize(si(P)-si(Q)+1);
    reverse(all(shou));
    
    auto hiku=atcoder::convolution(Q,shou);
    
    vector<mint> amari(si(P));
    for(int i=0;i<si(P);i++){
        amari[i]=P[i]-hiku[i];
    }
    while(si(shou)&&shou.back()==0) shou.pop_back();
    while(si(amari)&&amari.back()==0) amari.pop_back();
    return mp(shou,amari);
}
// 最高位が0でないようにしている(0のときは空)
// 多項式での除算

vector<mint> multieval(vector<mint> P,vector<mint> que){
    if(si(que)==0) return {};
    int N=si(que),n=1;
    while(n<N) n*=2;
    que.resize(n);
    
    vector<vector<mint>> Atree(n+n-1),Btree(n+n-1);
    for(int i=0;i<n;i++) Atree[n-1+i]={-que[i],1};
    for(int i=n-2;i>=0;i--){
        Atree[i]=atcoder::convolution(Atree[2*i+1],Atree[2*i+2]);
    }
    
    Btree[0]=warizan(P,Atree[0]).se;
    for(int i=1;i<n+n-1;i++){
        Btree[i]=warizan(Btree[(i-1)/2],Atree[i]).se;
    }
    
    vector<mint> res(N,0);
    for(int i=0;i<N;i++){
        if(si(Btree[n-1+i])) res[i]=Btree[n-1+i][0];
    }
    
    return res;
}

vector<mint> multieval_touhi(vector<mint> P,mint w,int M){
    if(M==0) return {};
    
    int N=si(P);
    
    if(N==0) return vector<mint>(M,0);
    
    if(w==0){
        vector<mint> res(M,P[0]);
        res[0]=0;
        for(int i=0;i<N;i++) res[0]+=P[i];
        return res;
    }
    
    vector<mint> y(N),v(N+M-1);
    for(ll i=0;i<N;i++) y[i]=P[i]/w.pow(i*(i-1)/2);
    for(ll i=0;i<N+M-1;i++) v[i]=w.pow(i*(i-1)/2);
    
    reverse(all(y));
    
    auto z=atcoder::convolution(y,v);
    
    vector<mint> res(M);
    
    for(ll i=0;i<M;i++){
        res[i]=z[N-1+i]/w.pow(i*(i-1)/2);
    }
    
    return res;
}
// w^0,...,w^(M-1)まで答える
// 0^0=1

int solve(int a,int b,int c,int d){
    int ans=0;
    for(int x=a;x<=17;x++){
        for(int y=x;y<=17;y++){
            for(int z=y;z<=17;z++){
                for(int s=b;s<=17;s++){
                    for(int t=s;t<=17;t++){
                        for(int u=c;u<=17;u++){
                            bool f=true;
                            f&=(x<=b);
                            f&=(y<=s&&s<=c);
                            f&=(z<=t&&t<=u&&u<=d);
                            
                            if(f) ans++;
                        }
                    }
                }
            }
        }
    }
    
    return ans;
}

int solve(int a,int b,int c,int d,int e){
    int ans=0;
    for(int x=a;x<=7;x++){
        for(int y=x;y<=7;y++){
            for(int z=y;z<=7;z++){
                for(int w=z;w<=7;w++){
                    for(int s=b;s<=7;s++){
                        for(int t=s;t<=7;t++){
                            for(int u=t;u<=7;u++){
                                for(int n=c;n<=7;n++){
                                    for(int m=n;m<=7;m++){
                                        for(int l=d;l<=7;l++){
                                            bool f=true;
                                            f&=(x<=b);
                                            f&=(y<=s&&s<=c);
                                            f&=(z<=t&&t<=n&&n<=d);
                                            f&=(w<=u&&u<=m&&m<=l&&l<=e);
                                            
                                            if(f) ans++;
                                        }
                                    }
                                }
                            }
                        }
                    }
                }
                
            }
        }
    }
    
    return ans;
}

int main(){
    
    std::ifstream in("text.txt");
    std::cin.rdbuf(in.rdbuf());
    cin.tie(0);
    ios::sync_with_stdio(false);
    
    ll N;cin>>N;
    vector<ll> A(N);
    for(int i=0;i<N;i++){
        cin>>A[i];A[i]+=i;
    }
    
    make();
    
    deque<vector<mint>> deq;
    for(int i=0;i<N;i++){
        deq.push_back({-A[i],1});
    }
    while(si(deq)>1){
        auto a=deq.front();deq.pop_front();
        auto b=deq.front();deq.pop_front();
        deq.push_back(atcoder::convolution(a,b));
    }
    
    vector<mint> que;
    for(int i=0;i<N;i++) que.push_back(A[i]);
    auto anss=multieval(bibun(deq[0],si(deq[0])+5),que);
    
    mint ans=1;
    for(mint x:anss) ans*=x;
    
    for(int i=0;i<N;i++){
        ans*=finv[i];
        ans*=finv[i];
    }
    
    if((N*(N-1)/2)&1) ans*=-1;
    cout<<ans.val()<<endl;
    
    return 0;
    
    for(int a=1;a<=17;a++){
        for(int b=a;b<=17;b++){
            for(int c=b;c<=17;c++){
                for(int d=c;d<=17;d++){
                    ll re=solve(a,b,c,d);
                    
                    vector<ll> A={a,b,c,d};
                    
                    mint ans=1;
                    
                    for(int i=0;i<si(A);i++){
                        for(int j=i+1;j<si(A);j++){
                            ans*=(A[j]-A[i]+j-i);
                        }
                        ans*=finv[i];
                        //cout<<ans.val()<<endl;
                    }
                    cout<<a<<" "<<b<<" "<<c<<" "<<d<<" "<<solve(a,b,c,d)<<" "<<ans.val()<<endl;
                    assert(ans.val()==re);
                }
            }
        }
    }
    return 0;
    for(int a=1;a<=7;a++){
        for(int b=a;b<=7;b++){
            for(int c=b;c<=7;c++){
                for(int d=c;d<=7;d++){
                    for(int e=d;e<=7;e++){
                        ll re=solve(a,b,c,d,e);
                        
                        vector<ll> A={a,b,c,d,e};
                        
                        mint ans=1;
                        
                        for(int i=0;i<si(A);i++){
                            for(int j=i+1;j<si(A);j++){
                                ans*=(A[j]-A[i]+j-i);
                            }
                            ans*=finv[i];
                            //cout<<ans.val()<<endl;
                        }
                        cout<<a<<" "<<b<<" "<<c<<" "<<d<<" "<<e<<" "<<solve(a,b,c,d,e)<<" "<<ans.val()<<endl;
                        assert(ans.val()==re);
                    }
                    //cout<<a<<" "<<b<<" "<<c<<" "<<d<<" "<<solve(a,b,c,d)<<endl;
                    
                }
            }
        }
    }
}