#include #include #include #include #include #include #include #include #include #include #include #include #include #include #include #include #include #include #include #include #include #include #include #include #include #include #include #include #include #include using namespace std; using lint = long long; using pint = pair; using plint = pair; struct fast_ios { fast_ios(){ cin.tie(nullptr), ios::sync_with_stdio(false), cout << fixed << setprecision(20); }; } fast_ios_; #define ALL(x) (x).begin(), (x).end() #define FOR(i, begin, end) for(int i=(begin),i##_end_=(end);i=i##_begin_;i--) #define REP(i, n) FOR(i,0,n) #define IREP(i, n) IFOR(i,0,n) template void ndarray(vector& vec, const V& val, int len) { vec.assign(len, val); } template void ndarray(vector& vec, const V& val, int len, Args... args) { vec.resize(len), for_each(begin(vec), end(vec), [&](T& v) { ndarray(v, val, args...); }); } template bool chmax(T &m, const T q) { return m < q ? (m = q, true) : false; } template bool chmin(T &m, const T q) { return m > q ? (m = q, true) : false; } int floor_lg(long long x) { return x <= 0 ? -1 : 63 - __builtin_clzll(x); } template pair operator+(const pair &l, const pair &r) { return make_pair(l.first + r.first, l.second + r.second); } template pair operator-(const pair &l, const pair &r) { return make_pair(l.first - r.first, l.second - r.second); } template vector sort_unique(vector vec) { sort(vec.begin(), vec.end()), vec.erase(unique(vec.begin(), vec.end()), vec.end()); return vec; } template int arglb(const std::vector &v, const T &x) { return std::distance(v.begin(), std::lower_bound(v.begin(), v.end(), x)); } template int argub(const std::vector &v, const T &x) { return std::distance(v.begin(), std::upper_bound(v.begin(), v.end(), x)); } template istream &operator>>(istream &is, vector &vec) { for (auto &v : vec) is >> v; return is; } template ostream &operator<<(ostream &os, const vector &vec) { os << '['; for (auto v : vec) os << v << ','; os << ']'; return os; } template ostream &operator<<(ostream &os, const array &arr) { os << '['; for (auto v : arr) os << v << ','; os << ']'; return os; } #if __cplusplus >= 201703L template istream &operator>>(istream &is, tuple &tpl) { std::apply([&is](auto &&... args) { ((is >> args), ...);}, tpl); return is; } template ostream &operator<<(ostream &os, const tuple &tpl) { os << '('; std::apply([&os](auto &&... args) { ((os << args << ','), ...);}, tpl); return os << ')'; } #endif template ostream &operator<<(ostream &os, const deque &vec) { os << "deq["; for (auto v : vec) os << v << ','; os << ']'; return os; } template ostream &operator<<(ostream &os, const set &vec) { os << '{'; for (auto v : vec) os << v << ','; os << '}'; return os; } template ostream &operator<<(ostream &os, const unordered_set &vec) { os << '{'; for (auto v : vec) os << v << ','; os << '}'; return os; } template ostream &operator<<(ostream &os, const multiset &vec) { os << '{'; for (auto v : vec) os << v << ','; os << '}'; return os; } template ostream &operator<<(ostream &os, const unordered_multiset &vec) { os << '{'; for (auto v : vec) os << v << ','; os << '}'; return os; } template ostream &operator<<(ostream &os, const pair &pa) { os << '(' << pa.first << ',' << pa.second << ')'; return os; } template ostream &operator<<(ostream &os, const map &mp) { os << '{'; for (auto v : mp) os << v.first << "=>" << v.second << ','; os << '}'; return os; } template ostream &operator<<(ostream &os, const unordered_map &mp) { os << '{'; for (auto v : mp) os << v.first << "=>" << v.second << ','; os << '}'; return os; } #ifdef HITONANODE_LOCAL const string COLOR_RESET = "\033[0m", BRIGHT_GREEN = "\033[1;32m", BRIGHT_RED = "\033[1;31m", BRIGHT_CYAN = "\033[1;36m", NORMAL_CROSSED = "\033[0;9;37m", RED_BACKGROUND = "\033[1;41m", NORMAL_FAINT = "\033[0;2m"; #define dbg(x) cerr << BRIGHT_CYAN << #x << COLOR_RESET << " = " << (x) << NORMAL_FAINT << " (L" << __LINE__ << ") " << __FILE__ << COLOR_RESET << endl #define dbgif(cond, x) ((cond) ? cerr << BRIGHT_CYAN << #x << COLOR_RESET << " = " << (x) << NORMAL_FAINT << " (L" << __LINE__ << ") " << __FILE__ << COLOR_RESET << endl : cerr) #else #define dbg(x) (x) #define dbgif(cond, x) 0 #endif template struct ModInt { #if __cplusplus >= 201402L #define MDCONST constexpr #else #define MDCONST #endif using lint = long long; MDCONST static int mod() { return md; } static int get_primitive_root() { static int primitive_root = 0; if (!primitive_root) { primitive_root = [&]() { std::set fac; int v = md - 1; for (lint i = 2; i * i <= v; i++) while (v % i == 0) fac.insert(i), v /= i; if (v > 1) fac.insert(v); for (int g = 1; g < md; g++) { bool ok = true; for (auto i : fac) if (ModInt(g).pow((md - 1) / i) == 1) { ok = false; break; } if (ok) return g; } return -1; }(); } return primitive_root; } int val; MDCONST ModInt() : val(0) {} MDCONST ModInt &_setval(lint v) { return val = (v >= md ? v - md : v), *this; } MDCONST ModInt(lint v) { _setval(v % md + md); } MDCONST explicit operator bool() const { return val != 0; } MDCONST ModInt operator+(const ModInt &x) const { return ModInt()._setval((lint)val + x.val); } MDCONST ModInt operator-(const ModInt &x) const { return ModInt()._setval((lint)val - x.val + md); } MDCONST ModInt operator*(const ModInt &x) const { return ModInt()._setval((lint)val * x.val % md); } MDCONST ModInt operator/(const ModInt &x) const { return ModInt()._setval((lint)val * x.inv() % md); } MDCONST ModInt operator-() const { return ModInt()._setval(md - val); } MDCONST ModInt &operator+=(const ModInt &x) { return *this = *this + x; } MDCONST ModInt &operator-=(const ModInt &x) { return *this = *this - x; } MDCONST ModInt &operator*=(const ModInt &x) { return *this = *this * x; } MDCONST ModInt &operator/=(const ModInt &x) { return *this = *this / x; } friend MDCONST ModInt operator+(lint a, const ModInt &x) { return ModInt()._setval(a % md + x.val); } friend MDCONST ModInt operator-(lint a, const ModInt &x) { return ModInt()._setval(a % md - x.val + md); } friend MDCONST ModInt operator*(lint a, const ModInt &x) { return ModInt()._setval(a % md * x.val % md); } friend MDCONST ModInt operator/(lint a, const ModInt &x) { return ModInt()._setval(a % md * x.inv() % md); } MDCONST bool operator==(const ModInt &x) const { return val == x.val; } MDCONST bool operator!=(const ModInt &x) const { return val != x.val; } MDCONST bool operator<(const ModInt &x) const { return val < x.val; } // To use std::map friend std::istream &operator>>(std::istream &is, ModInt &x) { lint t; return is >> t, x = ModInt(t), is; } MDCONST friend std::ostream &operator<<(std::ostream &os, const ModInt &x) { return os << x.val; } MDCONST ModInt pow(lint n) const { ModInt ans = 1, tmp = *this; while (n) { if (n & 1) ans *= tmp; tmp *= tmp, n >>= 1; } return ans; } static std::vector facs, facinvs, invs; MDCONST static void _precalculation(int N) { int l0 = facs.size(); if (N > md) N = md; if (N <= l0) return; facs.resize(N), facinvs.resize(N), invs.resize(N); for (int i = l0; i < N; i++) facs[i] = facs[i - 1] * i; facinvs[N - 1] = facs.back().pow(md - 2); for (int i = N - 2; i >= l0; i--) facinvs[i] = facinvs[i + 1] * (i + 1); for (int i = N - 1; i >= l0; i--) invs[i] = facinvs[i] * facs[i - 1]; } MDCONST lint inv() const { if (this->val < std::min(md >> 1, 1 << 21)) { while (this->val >= int(facs.size())) _precalculation(facs.size() * 2); return invs[this->val].val; } else { return this->pow(md - 2).val; } } MDCONST ModInt fac() const { while (this->val >= int(facs.size())) _precalculation(facs.size() * 2); return facs[this->val]; } MDCONST ModInt facinv() const { while (this->val >= int(facs.size())) _precalculation(facs.size() * 2); return facinvs[this->val]; } MDCONST ModInt doublefac() const { lint k = (this->val + 1) / 2; return (this->val & 1) ? ModInt(k * 2).fac() / (ModInt(2).pow(k) * ModInt(k).fac()) : ModInt(k).fac() * ModInt(2).pow(k); } MDCONST ModInt nCr(const ModInt &r) const { return (this->val < r.val) ? 0 : this->fac() * (*this - r).facinv() * r.facinv(); } MDCONST ModInt nPr(const ModInt &r) const { return (this->val < r.val) ? 0 : this->fac() * (*this - r).facinv(); } ModInt sqrt() const { if (val == 0) return 0; if (md == 2) return val; if (pow((md - 1) / 2) != 1) return 0; ModInt b = 1; while (b.pow((md - 1) / 2) == 1) b += 1; int e = 0, m = md - 1; while (m % 2 == 0) m >>= 1, e++; ModInt x = pow((m - 1) / 2), y = (*this) * x * x; x *= (*this); ModInt z = b.pow(m); while (y != 1) { int j = 0; ModInt t = y; while (t != 1) j++, t *= t; z = z.pow(1LL << (e - j - 1)); x *= z, z *= z, y *= z; e = j; } return ModInt(std::min(x.val, md - x.val)); } }; template std::vector> ModInt::facs = {1}; template std::vector> ModInt::facinvs = {1}; template std::vector> ModInt::invs = {0}; using mint = ModInt<998244353>; template struct matrix { int H, W; std::vector elem; typename std::vector::iterator operator[](int i) { return elem.begin() + i * W; } inline T &at(int i, int j) { return elem[i * W + j]; } inline T get(int i, int j) const { return elem[i * W + j]; } int height() const { return H; } int width() const { return W; } std::vector> vecvec() const { std::vector> ret(H); for (int i = 0; i < H; i++) { std::copy(elem.begin() + i * W, elem.begin() + (i + 1) * W, std::back_inserter(ret[i])); } return ret; } operator std::vector>() const { return vecvec(); } matrix() = default; matrix(int H, int W) : H(H), W(W), elem(H * W) {} matrix(const std::vector> &d) : H(d.size()), W(d.size() ? d[0].size() : 0) { for (auto &raw : d) std::copy(raw.begin(), raw.end(), std::back_inserter(elem)); } static matrix Identity(int N) { matrix ret(N, N); for (int i = 0; i < N; i++) ret.at(i, i) = 1; return ret; } matrix operator-() const { matrix ret(H, W); for (int i = 0; i < H * W; i++) ret.elem[i] = -elem[i]; return ret; } matrix operator*(const T &v) const { matrix ret = *this; for (auto &x : ret.elem) x *= v; return ret; } matrix operator/(const T &v) const { matrix ret = *this; const T vinv = T(1) / v; for (auto &x : ret.elem) x *= vinv; return ret; } matrix operator+(const matrix &r) const { matrix ret = *this; for (int i = 0; i < H * W; i++) ret.elem[i] += r.elem[i]; return ret; } matrix operator-(const matrix &r) const { matrix ret = *this; for (int i = 0; i < H * W; i++) ret.elem[i] -= r.elem[i]; return ret; } matrix operator*(const matrix &r) const { matrix ret(H, r.W); for (int i = 0; i < H; i++) { for (int k = 0; k < W; k++) { for (int j = 0; j < r.W; j++) ret.at(i, j) += this->get(i, k) * r.get(k, j); } } return ret; } matrix &operator*=(const T &v) { return *this = *this * v; } matrix &operator/=(const T &v) { return *this = *this / v; } matrix &operator+=(const matrix &r) { return *this = *this + r; } matrix &operator-=(const matrix &r) { return *this = *this - r; } matrix &operator*=(const matrix &r) { return *this = *this * r; } bool operator==(const matrix &r) const { return H == r.H and W == r.W and elem == r.elem; } bool operator!=(const matrix &r) const { return H != r.H or W != r.W or elem != r.elem; } bool operator<(const matrix &r) const { return elem < r.elem; } matrix pow(int64_t n) const { matrix ret = Identity(H); bool ret_is_id = true; if (n == 0) return ret; for (int i = 63 - __builtin_clzll(n); i >= 0; i--) { if (!ret_is_id) ret *= ret; if ((n >> i) & 1) ret *= (*this), ret_is_id = false; } return ret; } std::vector pow_vec(int64_t n, std::vector vec) const { matrix x = *this; while (n) { if (n & 1) vec = x * vec; x *= x; n >>= 1; } return vec; }; matrix transpose() const { matrix ret(W, H); for (int i = 0; i < H; i++) { for (int j = 0; j < W; j++) ret.at(j, i) = this->get(i, j); } return ret; } // Gauss-Jordan elimination // - Require inverse for every non-zero element // - Complexity: O(H^2 W) template ::value>::type * = nullptr> static int choose_pivot(const matrix &mtr, int h, int c) noexcept { int piv = -1; for (int j = h; j < mtr.H; j++) { if (mtr.get(j, c) and (piv < 0 or std::abs(mtr.get(j, c)) > std::abs(mtr.get(piv, c)))) piv = j; } return piv; } template ::value>::type * = nullptr> static int choose_pivot(const matrix &mtr, int h, int c) noexcept { for (int j = h; j < mtr.H; j++) { if (mtr.get(j, c)) return j; } return -1; } matrix gauss_jordan() const { int c = 0; matrix mtr(*this); std::vector ws; ws.reserve(W); for (int h = 0; h < H; h++) { if (c == W) break; int piv = choose_pivot(mtr, h, c); if (piv == -1) { c++; h--; continue; } if (h != piv) { for (int w = 0; w < W; w++) { std::swap(mtr[piv][w], mtr[h][w]); mtr.at(piv, w) *= -1; // To preserve sign of determinant } } ws.clear(); for (int w = c; w < W; w++) { if (mtr.at(h, w) != 0) ws.emplace_back(w); } const T hcinv = T(1) / mtr.at(h, c); for (int hh = 0; hh < H; hh++) if (hh != h) { const T coeff = mtr.at(hh, c) * hcinv; for (auto w : ws) mtr.at(hh, w) -= mtr.at(h, w) * coeff; mtr.at(hh, c) = 0; } c++; } return mtr; } int rank_of_gauss_jordan() const { for (int i = H * W - 1; i >= 0; i--) { if (elem[i]) return i / W + 1; } return 0; } T determinant_of_upper_triangle() const { T ret = 1; for (int i = 0; i < H; i++) ret *= get(i, i); return ret; } int inverse() { assert(H == W); std::vector> ret = Identity(H), tmp = *this; int rank = 0; for (int i = 0; i < H; i++) { int ti = i; while (ti < H and tmp[ti][i] == 0) ti++; if (ti == H) { continue; } else { rank++; } ret[i].swap(ret[ti]), tmp[i].swap(tmp[ti]); T inv = T(1) / tmp[i][i]; for (int j = 0; j < W; j++) ret[i][j] *= inv; for (int j = i + 1; j < W; j++) tmp[i][j] *= inv; for (int h = 0; h < H; h++) { if (i == h) continue; const T c = -tmp[h][i]; for (int j = 0; j < W; j++) ret[h][j] += ret[i][j] * c; for (int j = i + 1; j < W; j++) tmp[h][j] += tmp[i][j] * c; } } *this = ret; return rank; } friend std::vector operator*(const matrix &m, const std::vector &v) { assert(m.W == int(v.size())); std::vector ret(m.H); for (int i = 0; i < m.H; i++) { for (int j = 0; j < m.W; j++) ret[i] += m.get(i, j) * v[j]; } return ret; } friend std::vector operator*(const std::vector &v, const matrix &m) { assert(int(v.size()) == m.H); std::vector ret(m.W); for (int i = 0; i < m.H; i++) { for (int j = 0; j < m.W; j++) ret[j] += v[i] * m.get(i, j); } return ret; } std::vector prod(const std::vector &v) const { return (*this) * v; } std::vector prod_left(const std::vector &v) const { return v * (*this); } friend std::ostream &operator<<(std::ostream &os, const matrix &x) { os << "[(" << x.H << " * " << x.W << " matrix)"; os << "\n[column sums: "; for (int j = 0; j < x.W; j++) { T s = 0; for (int i = 0; i < x.H; i++) s += x.get(i, j); os << s << ","; } os << "]"; for (int i = 0; i < x.H; i++) { os << "\n["; for (int j = 0; j < x.W; j++) os << x.get(i, j) << ","; os << "]"; } os << "]\n"; return os; } friend std::istream &operator>>(std::istream &is, matrix &x) { for (auto &v : x.elem) is >> v; return is; } }; int main() { int N, Q; cin >> N >> Q; mint ret = 0; const mint pall = mint(N) * (N + 1); FOR(w, 1, N) { // A[i], A[j] を見る j - i = w mint ptr = mint(w) * (N + 1 - w); mint q = pall - ptr; matrix mat(3, 3); mat[0][0] = mat[1][1] = q; mat[0][1] = mat[1][0] = ptr; mat[2][2] = pall; mint coeff = mint(N + 1 - w) * (N - w) * (N - w + 1) / 2; coeff -= mint(N - w) * (N - w + 1) * (2 * (N - w) + 1) / 6; mat[2][1] = coeff; ret += mat.pow_vec(Q, vector{1, 0, 0}).at(2); } cout << ret / 2 << endl; }