#line 1 ".lib/template.hpp" #include #include #include #include #include #include #include #include #include #include #include #include #include #include #include #include #include #include #include #include #include #include #include #include #define allof(obj) (obj).begin(), (obj).end() #define range(i, l, r) for(int i=l;i>1)|y_bit)) #define bit_kth(i, k) ((i >> k)&1) #define bit_highest(i) (i?63-__builtin_clzll(i):-1) #define bit_lowest(i) (i?__builtin_ctzll(i):-1) #define sleepms(t) std::this_thread::sleep_for(std::chrono::milliseconds(t)) using ll = long long; using ld = long double; using ul = uint64_t; using pi = std::pair; using pl = std::pair; using namespace std; template std::ostream &operator<<(std::ostream &dest, const std::pair &p){ dest << p.first << ' ' << p.second; return dest; } template std::ostream &operator<<(std::ostream &dest, const std::vector> &v){ int sz = v.size(); if(sz==0) return dest; for(int i=0;i std::ostream &operator<<(std::ostream &dest, const std::vector &v){ int sz = v.size(); if(sz==0) return dest; for(int i=0;i std::ostream &operator<<(std::ostream &dest, const std::array &v){ if(sz==0) return dest; for(int i=0;i std::ostream &operator<<(std::ostream &dest, const std::set &v){ for(auto itr=v.begin();itr!=v.end();){ dest << *itr; itr++; if(itr!=v.end()) dest << ' '; } return dest; } template std::ostream &operator<<(std::ostream &dest, const std::map &v){ for(auto itr=v.begin();itr!=v.end();){ dest << '(' << itr->first << ", " << itr->second << ')'; itr++; if(itr!=v.end()) dest << '\n'; } return dest; } template vector make_vec(size_t sz, T val){return std::vector(sz, val);} template auto make_vec(size_t sz, Tail ...tail){ return std::vector(tail...))>(sz, make_vec(tail...)); } template vector read_vec(size_t sz){ std::vector v(sz); for(int i=0;i<(int)sz;i++) std::cin >> v[i]; return v; } template auto read_vec(size_t sz, Tail ...tail){ auto v = std::vector(tail...))>(sz); for(int i=0;i<(int)sz;i++) v[i] = read_vec(tail...); return v; } void io_init(){ std::cin.tie(nullptr); std::ios::sync_with_stdio(false); } #line 1 ".lib/math/mod.hpp" #line 7 ".lib/math/mod.hpp" #include #line 9 ".lib/math/mod.hpp" #include #line 13 ".lib/math/mod.hpp" // @param m `1 <= m` // @return x mod m 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; explicit 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 long long y = x * _m; return (unsigned int)(z - y + (z < y ? _m : 0)); } }; // @param n `0 <= n` // @param m `1 <= m` // @return `(x ** n) % m` 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; } // Reference: // M. Forisek and J. Jancina, // Fast Primality Testing for Integers That Fit into a Machine Word // @param n `0 <= n` 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; constexpr long long bases[3] = {2, 7, 61}; for (long long a : bases) { 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 constexpr bool is_prime = is_prime_constexpr(n); 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 constexpr int primitive_root = primitive_root_constexpr(m); int ceil_pow2(int n) { int x = 0; while ((1U << x) < (unsigned int)(n)) x++; return x; } int bsf(unsigned int n) { return __builtin_ctz(n); } // @param b `1 <= b` // @return pair(g, x) s.t. g = gcd(a, b), xa = g (mod b), 0 <= x < b/g constexpr std::pair 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; auto tmp = s; s = t; t = tmp; tmp = m0; m0 = m1; m1 = tmp; } if (m0 < 0) m0 += b / s; return {s, m0}; } struct modint_base {}; struct static_modint_base : modint_base {}; template using is_modint = std::is_base_of; template using is_modint_t = std::enable_if_t::value>; template * = nullptr> struct static_modint : 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 static_modint(T v){ long long x = (long long)(v % (long long)(umod())); if (x < 0) x += umod(); _v = (unsigned int)(x); } 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 = 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 = is_prime; }; template struct dynamic_modint : modint_base { using mint = dynamic_modint; public: static int mod() { return (int)(bt.umod()); } static void set_mod(int m) { assert(1 <= m); bt = barrett(m); } static mint raw(int v) { mint x; x._v = v; return x; } dynamic_modint() : _v(0) {} template dynamic_modint(T v) { long long x = (long long)(v % (long long)(mod())); if (x < 0) x += mod(); _v = (unsigned int)(x); } 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 = 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 barrett bt; static unsigned int umod() { return bt.umod(); } }; template barrett dynamic_modint::bt(998244353); using modint998244353 = static_modint<998244353>; using modint1000000007 = static_modint<1000000007>; using modint = dynamic_modint<-1>; template using is_static_modint = std::is_base_of; template using is_static_modint_t = std::enable_if_t::value>; template struct is_dynamic_modint : public std::false_type {}; template struct is_dynamic_modint> : public std::true_type {}; template using is_dynamic_modint_t = std::enable_if_t::value>; template std::ostream &operator<<(std::ostream &dest, const static_modint &a){ dest << a.val(); return dest; } template std::ostream &operator<<(std::ostream &dest, const dynamic_modint &a){ dest << a.val(); return dest; } // 0 <= n < m <= int_max // 前処理 O(n + log(m)) // 各種計算 O(1) // 変数 <= n #line 406 ".lib/math/mod.hpp" template struct modcomb{ private: int n; std::vector f, i, fi; void init(int _n){ assert(0 <= _n && _n < mint::mod()); if(_n < f.size()) return; n = _n; f.resize(n + 1), i.resize(n + 1), fi.resize(n + 1); f[0] = fi[0] = mint(1); if(n) f[1] = fi[1] = i[1] = mint(1); for(int j = 2; j <= n; j++) f[j] = f[j - 1] * j; fi[n] = f[n].inv(); for(int j = n; j >= 2; j--){ fi[j - 1] = fi[j] * j; i[j] = f[j - 1] * fi[j]; } } public: modcomb(): n(-1){} modcomb(int _n){ init(_n); } void recalc(int _n){ init(std::min(mint::mod() - 1, 1 << ceil_pow2(_n))); } mint comb(int a, int b){ if((a < 0) || (b < 0) || (a < b)) return 0; return f[a] * fi[a - b] * fi[b]; } mint perm(int a, int b){ if((a < 0) || (b < 0) || (a < b)) return 0; return f[a] * fi[a - b]; } mint fac(int x){ assert(0 <= x && x <= n); return f[x]; } mint inv(int x){ assert(0 < x && x <= n); return i[x]; } mint finv(int x){ assert(0 <= x && x <= n); return fi[x]; } }; // mod == 2: 定数時間 // modが素数: O(min(n, mod) + log(n)) template struct lucas_prime{ using mint = dynamic_modint; modcomb mcb; void set_mod(int mod){ mint::set_mod(mod); } int comb(long long n, long long r){ if(mint::mod() == 1 || n < 0 || r < 0 || n < r) return 0; if(mint::mod() == 2) return (n & r) == r; mcb.recalc(std::min(n, (long long)mint::mod())); mint res = 1; while(n){ int x = n % mint::mod(), y = r % mint::mod(); res *= mcb.comb(x, y); n /= mint::mod(), r /= mint::mod(); } return res.val(); } }; template struct modpow_table{ std::vector v; // x^maxkまで計算できる modpow_table(){} void init(int x, int maxk){ v.resize(maxk + 1); v[0] = 1; for(int i = 1; i <= maxk; i++) v[i] = v[i - 1] * x; } mint pow(int k){ assert(0 <= k && k < v.size()); return v[k]; } }; template int modpow(long long a, long long b){ int ret = (m == 1 ? 0 : 1), mul = a % m; while(b){ if(b & 1) ret = ((long long)ret * mul) % m; mul = ((long long)mul * mul) % m; b >>= 1; } return ret; } int modpow(long long a, long long b, int m){ int ret = (m == 1 ? 0 : 1), mul = a % m; while(b){ if(b & 1) ret = ((long long)ret * mul) % m; mul = ((long long)mul * mul) % m; b >>= 1; } return ret; } #line 514 ".lib/math/mod.hpp" // modpow(x,2,mod) == aとなるxを返す // 存在しないなら-1 // mod は素数 long long modsqrt(long long a, long long mod){ a %= mod; if(a == 0) return 0LL; if(mod == 2) return 1LL; if(modpow(a, (mod - 1) / 2, mod) != 1) return -1LL; if(mod % 4 == 3) return modpow(a, mod / 4 + 1, mod); long long q = mod - 1, m = 0; while(q % 2 == 0) q >>= 1, m++; std::mt19937 mt; long long z; do{ z = mt() % mod; }while(modpow(z, (mod - 1) / 2, mod) != mod - 1); long long c = modpow(z, q, mod); long long t = modpow(a, q, mod); long long r = modpow(a, (q + 1) >> 1, mod); for(; m > 1; --m) { long long tmp = modpow(t, 1LL << (m - 2), mod); if(tmp != 1) r = r * c % mod, t = t * (c * c % mod) % mod; c = c * c % mod; } return r; } //n次以下の多項式に対し //f(0) ~ f(n)を与えf(p)を求める //O(n log(MOD)) template mint __lagrange(const std::vector &y, mint p, modcomb &mcb){ int sz = y.size(); mcb.recalc(sz); mint M = 1, res = 0; std::vector itable(sz, 1), num(sz); if(p.val() < sz) return y[p.val()]; for(int i = 0; i < sz; i++){ M *= p - i; num[i] = p - i; } uint32_t cnt = mint::mod() - 2; while(cnt){ if(cnt & 1){ for(int i = 0; i < sz; i++) itable[i] *= num[i], num[i] *= num[i]; }else{ for(int i = 0; i < sz; i++) num[i] *= num[i]; } cnt >>= 1; } for(int i = 0; i < sz; i++){ mint iQ = mcb.finv(i) * mcb.finv(sz - 1 - i); if((sz - i - 1) & 1) iQ *= -1; res += y[i] * iQ * itable[i]; } return res * M; } template mint lagrange(const std::vector &y, mint p){ modcomb mcb; return __lagrange(y, p, mcb); } template mint riid(mint r, int d, long long n){ if(n == 0) return 0; if(r.val() == 0){ return d == 0 ? mint(1) : 0; } n--; std::vector y(d + 2), ipow(d + 2, 1), tbl(d + 2); for(int i = 0; i < d + 2; i++) tbl[i] = i; int cnt = d; while(cnt){ if(cnt & 1){ for(int i = 0; i < d + 2; i++) ipow[i] *= tbl[i], tbl[i] *= tbl[i]; }else{ for(int i = 0; i < d + 2; i++) tbl[i] *= tbl[i]; } cnt >>= 1; } mint tmp = 0, rpow = 1, last = r.pow(n % (mint::mod() - 1)); n %= mint::mod(); for(int i = 0; i < d + 2; i++){ tmp += rpow * ipow[i]; rpow *= r; y[i] = tmp; } modcomb mcb(y.size()); if(r.val() == 1) return __lagrange(y, n, mcb); ipow[0] = 1, ipow[1] = -r; mint comb = 1, c = 0; for(int i = 2; i < d + 2; i++) ipow[i] = ipow[i - 1] * ipow[1]; for(int i = 0; i < d + 1; i++){ comb *= mcb.inv(i + 1) * mint(d + 1 - i); c += comb * ipow[d - i] * y[i]; } mint di = mint(1 - r); c *= di.pow(d + 1).inv(); mint powerRinv = 1, rinv = r.inv(); for(int i = 0; i < d + 1; i++){ y[i] -= c; y[i] *= powerRinv; powerRinv *= rinv; } y.pop_back(); return c + last * __lagrange(y, n, mcb); } #line 3 "c.cpp" using mint = modint998244353; int main(){ io_init(); modcomb mcb(1000000); mint ans = 0; ll n, m; std::cin >> n >> m; if(m == 0){ if(n & 1){ std::cout << 0 << '\n'; }else{ for(int i = 0; i <= (n / 2); i++){ int j = (n - 2 * i) / 2; ans += mcb.comb(n, 2 * i) * mcb.comb(2 * i, i) * mcb.comb(n - 2 * i, j); } std::cout << ans * mint(4).pow(n) << '\n'; } return 0; } if(m < 0) m *= -1; for(ll i = 1; i * i <= m; i++){ if(m % i != 0) continue; ll j = m / i; if(i % 2 != j % 2) continue; // x - y = i && x + y = j // x = (j + i) / 2 // y = (j - i) / 2 ll x = (j + i) / 2; ll y = (j - i) / 2; if(x + y > n || (x + y) % 2 != n % 2) continue; mint tmp = 0; // (+-x, +-y) 4パターン // (x, y) を求めて × 4 for(ll yoko = x; yoko <= n - y; yoko += 2){ ll tate = n - yoko; //std::cout << mcb.comb(n, yoko) << " " << mcb.comb(yoko, x) << '\n'; tmp += mcb.comb(n, yoko) * mcb.comb(yoko, (yoko - x) / 2) * mcb.comb(tate, (tate - y) / 2); } if(y == 0) tmp *= 2; else tmp *= 4; ans += tmp; } std::cout << (ans / mint(4).pow(n)) << '\n'; }