#include #include using namespace std; #define rep(i,n) for(long long i = 0; i < (long long)(n); i++) #define repi(i,a,b) for(long long i = (long long)(a); i < (long long)(b); i++) #define pb push_back #define all(x) (x).begin(), (x).end() #define fi first #define se second #define mt make_tuple #define mp make_pair #define ZERO(a) memset(a,0,sizeof(a)) template bool chmin(T1 &a, T2 b) { return b < a && (a = b, true); } template bool chmax(T1 &a, T2 b) { return a < b && (a = b, true); } #define exists find_if #define forall all_of using ll = long long; using vll = vector; using vvll = vector; using P = pair; using ld = long double; using vld = vector; using vi = vector; using vvi = vector; vll conv(vi& v) { vll r(v.size()); rep(i, v.size()) r[i] = v[i]; return r; } inline void input(int &v){ v=0;char c=0;int p=1; while(c<'0' || c>'9'){if(c=='-')p=-1;c=getchar();} while(c>='0' && c<='9'){v=(v<<3)+(v<<1)+c-'0';c=getchar();} v*=p; } template ostream &operator<<(ostream &o, const pair &v) { o << "(" << v.first << ", " << v.second << ")"; return o; } template struct seq{}; template struct gen_seq : gen_seq{}; template struct gen_seq<0, Is...> : seq{}; template void print_tuple(basic_ostream& os, Tuple const& t, seq){ using s = int[]; (void)s{0, (void(os << (Is == 0? "" : ", ") << get(t)), 0)...}; } template auto operator<<(basic_ostream& os, tuple const& t) -> basic_ostream& { os << "("; print_tuple(os, t, gen_seq()); return os << ")"; } ostream &operator<<(ostream &o, const vvll &v) { rep(i, v.size()) { rep(j, v[i].size()) o << v[i][j] << " "; o << endl; } return o; } template ostream &operator<<(ostream &o, const vector &v) { o << '['; rep(i, v.size()) o << v[i] << (i != v.size()-1 ? ", " : ""); o << "]"; return o; } template ostream &operator<<(ostream &o, const set &m) { o << '['; for (auto it = m.begin(); it != m.end(); it++) o << *it << (next(it) != m.end() ? ", " : ""); o << "]"; return o; } template ostream &operator<<(ostream &o, const unordered_set &m) { o << '['; for (auto it = m.begin(); it != m.end(); it++) o << *it << (next(it) != m.end() ? ", " : ""); o << "]"; return o; } template ostream &operator<<(ostream &o, const map &m) { o << '['; for (auto it = m.begin(); it != m.end(); it++) o << *it << (next(it) != m.end() ? ", " : ""); o << "]"; return o; } template ostream &operator<<(ostream &o, const unordered_map &m) { o << '['; for (auto it = m.begin(); it != m.end(); it++) o << *it; o << "]"; return o; } vector range(const int x, const int y) { vector v(y - x + 1); iota(v.begin(), v.end(), x); return v; } template istream& operator>>(istream& i, vector& o) { rep(j, o.size()) i >> o[j]; return i;} string bits_to_string(ll input, ll n=64) { string s; rep(i, n) s += '0' + !!(input & (1ll << i)); reverse(all(s)); return s; } template ostream &operator<<(ostream &o, const priority_queue &v) { auto tmp = v; while (tmp.size()) { auto x = tmp.top(); tmp.pop(); o << x << " ";} o << endl; return o; } template unordered_map counter(vector vec){unordered_map ret; for (auto&& x : vec) ret[x]++; return ret;}; string substr(string s, P x) {return s.substr(x.fi, x.se - x.fi); } void vizGraph(vvll& g, int mode = 0, string filename = "out.png") { ofstream ofs("./out.dot"); ofs << "digraph graph_name {" << endl; set

memo; rep(i, g.size()) rep(j, g[i].size()) { if (mode && (memo.count(P(i, g[i][j])) || memo.count(P(g[i][j], i)))) continue; memo.insert(P(i, g[i][j])); ofs << " " << i << " -> " << g[i][j] << (mode ? " [arrowhead = none]" : "")<< endl; } ofs << "}" << endl; ofs.close(); system(((string)"dot -T png out.dot >" + filename).c_str()); } size_t random_seed; namespace std { using argument_type = P; template<> struct hash { size_t operator()(argument_type const& x) const { size_t seed = random_seed; seed ^= hash{}(x.fi); seed ^= (hash{}(x.se) << 1); return seed; } }; }; // hash for various class namespace myhash{ const int Bsizes[]={3,9,13,17,21,25,29,33,37,41,45,49,53,57,61,65,69,73,77,81}; const int xor_nums[]={0x100007d1,0x5ff049c9,0x14560859,0x07087fef,0x3e277d49,0x4dba1f17,0x709c5988,0x05904258,0x1aa71872,0x238819b3,0x7b002bb7,0x1cf91302,0x0012290a,0x1083576b,0x76473e49,0x3d86295b,0x20536814,0x08634f4d,0x115405e8,0x0e6359f2}; const int hash_key=xor_nums[rand()%20]; const int mod_key=xor_nums[rand()%20]; template struct myhash{ std::size_t operator()(const T& val) const { return (hash{}(val)%mod_key)^hash_key; } }; }; template class uset:public std::unordered_set> { using SET=std::unordered_set>; public: uset():SET(){SET::rehash(myhash::Bsizes[rand()%20]);} }; uint32_t randxor() { static uint32_t x=1+(uint32_t)random_seed,y=362436069,z=521288629,w=88675123; uint32_t t; t=(x^(x<<11));x=y;y=z;z=w; return( w=(w^(w>>19))^(t^(t>>8)) ); } struct timeval start; double sec() { struct timeval tv; gettimeofday(&tv, NULL); return (tv.tv_sec - start.tv_sec) + (tv.tv_usec - start.tv_usec) * 1e-6; } struct init_{init_(){ gettimeofday(&start, NULL); ios::sync_with_stdio(false); cin.tie(0); struct timeval myTime; struct tm *time_st; gettimeofday(&myTime, NULL); time_st = localtime(&myTime.tv_sec); srand(myTime.tv_usec); random_seed = RAND_MAX / 2 + rand() / 2; }} init__; #define rand randxor static const double EPS = 1e-14; static const long long INF = 1e18; static const long long mo = 1e9+7; #define ldout fixed << setprecision(40) class Mod { public: int num; Mod() : Mod(0) {} Mod(long long int n) : num(n) { } Mod(const string &s){ long long int tmp = 0; for(auto &c:s) tmp = (c-'0'+tmp*10) % mo; num = tmp; } Mod(int n) : Mod(static_cast(n)) {} operator int() { return num; } }; istream &operator>>(istream &is, Mod &x) { long long int n; is >> n; x = n; return is; } ostream &operator<<(ostream &o, const Mod &x) { o << x.num; return o; } Mod operator+(const Mod a, const Mod b) { return Mod((a.num + b.num) % mo); } Mod operator+(const long long int a, const Mod b) { return Mod(a) + b; } Mod operator+(const Mod a, const long long int b) { return b + a; } Mod operator++(Mod &a) { return a + Mod(1); } Mod operator-(const Mod a, const Mod b) { return Mod((mo + a.num - b.num) % mo); } Mod operator-(const long long int a, const Mod b) { return Mod(a) - b; } Mod operator--(Mod &a) { return a - Mod(1); } Mod operator*(const Mod a, const Mod b) { return Mod(((long long)a.num * b.num) % mo); } Mod operator*(const long long int a, const Mod b) { return Mod(a)*b; } Mod operator*(const Mod a, const long long int b) { return Mod(b)*a; } Mod operator*(const Mod a, const int b) { return Mod(b)*a; } Mod operator+=(Mod &a, const Mod b) { return a = a + b; } Mod operator+=(long long int &a, const Mod b) { return a = a + b; } Mod operator-=(Mod &a, const Mod b) { return a = a - b; } Mod operator-=(long long int &a, const Mod b) { return a = a - b; } Mod operator*=(Mod &a, const Mod b) { return a = a * b; } Mod operator*=(long long int &a, const Mod b) { return a = a * b; } Mod operator*=(Mod& a, const long long int &b) { return a = a * b; } Mod factorial(const long long n) { if (n < 0) return 0; Mod ret = 1; for (int i = 1; i <= n; i++) { ret *= i; } return ret; } Mod operator^(const Mod a, const long long n) { if (n == 0) return Mod(1); Mod res = (a * a) ^ (n / 2); if (n % 2) res = res * a; return res; } Mod modpowsum(const Mod a, const long long b) { if (b == 0) return 0; if (b % 2 == 1) return modpowsum(a, b - 1) * a + Mod(1); Mod result = modpowsum(a, b / 2); return result * (a ^ (b / 2)) + result; } /*************************************/ // 以下、modは素数でなくてはならない! /*************************************/ Mod inv(const Mod a) { return a ^ (mo - 2); } Mod operator/(const Mod a, const Mod b) { assert(b.num != 0); return a * inv(b); } Mod operator/(const long long int a, const Mod b) { assert(b.num != 0); return Mod(a) * inv(b); } Mod operator/=(Mod &a, const Mod b) { assert(b.num != 0); return a = a * inv(b); } // n!と1/n!のテーブルを作る。 // nCrを高速に計算するためのもの。 // // O(n + log mo) vector fact, rfact; void constructFactorial(const long long n) { fact.resize(n); rfact.resize(n); fact[0] = rfact[0] = 1; for (int i = 0; i < n - 1; i++) { fact[i+1] = fact[i] * (i+1); } rfact[n-1] = Mod(1) / fact[n-1]; for (int i = n - 1; i >= 1; i--) rfact[i-1] = rfact[i] * i; // ((n-1)!)^-1 = (n!)^-1 * n } // O(1) // constructFactorialしておけば、n, r=1e7くらいまではいけます Mod nCr(const long long n, const long long r) { if (n < 0 || r < 0) return 0; return fact[n] * rfact[r] * rfact[n-r]; } // O(r.size()) // sum(r)! / r[0]! / r[1]! / ... Mod nCr(const vector r) { ll sum = accumulate(all(r), 0ll); Mod ret = fact[sum]; rep(i, r.size()) ret *= rfact[r[i]]; return ret; } // O(k log mo) Mod nCrWithoutConstruction(const long long n, const long long k) { if (n < 0) return 0; if (k < 0) return 0; Mod ret = 1; for (int i = 0; i < k; i++) { ret *= n - (Mod)i; ret /= Mod(i+1); } return ret; } // n*mの盤面を左下から右上に行く場合の数 // O(1) Mod nBm(const long long n, const long long m) { if (n < 0 || m < 0) return 0; return nCr(n + m, n); } /*************************************/ // GF(p)の行列演算 /*************************************/ using number = Mod; using arr = vector; using matrix = vector>; ostream &operator<<(ostream &o, const arr &v) { rep(i, v.size()) cout << v[i] << " "; return o; } ostream &operator<<(ostream &o, const matrix &v) { rep(i, v.size()) cout << v[i]; return o; } matrix zero(int n) { return matrix(n, arr(n, 0)); } // O(n^2) matrix identity(int n) { matrix A(n, arr(n, 0)); rep(i, n) A[i][i] = 1; return A; } // O(n^2) // O(n^2) arr mul(const matrix &A, const arr &x) { arr y(A.size(), 0); rep(i, A.size()) rep(j, A[0].size()) y[i] += A[i][j] * x[j]; return y; } // O(n^3) matrix mul(const matrix &A, const matrix &B) { matrix C(A.size(), arr(B[0].size(), 0)); rep(i, C.size()) rep(j, C[i].size()) rep(k, A[i].size()) C[i][j] += A[i][k] * B[k][j]; return C; } // O(n^2) matrix plu(const matrix &A, const matrix &B) { matrix C(A.size(), arr(B[0].size(), 0)); rep(i, C.size()) rep(j, C[i].size()) C[i][j] += A[i][j] + B[i][j]; return C; } // O(n^2) arr plu(const arr &A, const arr &B) { arr C(A.size()); rep(i, A.size()) C[i] += A[i] + B[i]; return C; } // 構築なし累乗 // return A^e // // O(n^3 log e) matrix pow(const matrix &A, long long e) { return e == 0 ? identity(A.size()) : e % 2 == 0 ? pow(mul(A, A), e/2) : mul(A, pow(A, e-1)); } // 構築付き累乗 // // return powA: A^2^i // O(n^3 log e) matrix pow(const vector& powA, long long e) { // powA[0]がA // cout << powA[0] << "^" << e <& powA, long long e, arr& a) { // powA[0]がA // cout << powA[0] << "^" << e <& powA) { powA.clear(); powA.pb(A); for (int i = 1; (1ll << i) < e; i++) { powA.pb(mul(powA[i-1], powA[i-1])); } } // O(n) number inner_product(const arr &a, const arr &b) { number ans = 0; for (int i = 0; i < (int)a.size(); ++i) ans += a[i] * b[i]; return ans; } // O(n) number tr(const matrix &A) { number ans = 0; for (int i = 0; i < (int)A.size(); ++i) ans += A[i][i]; return ans; } // O( n^3 ) // modは素数でなければならない!! number det(matrix A) { int n = A.size(); assert(n == (int)A[0].size()); number ans = 1; for (int i = 0; i < n; i++) { int pivot = -1; for (int j = i; j < n; j++) if (A[j][i]) { pivot = j; break; } if (pivot == -1) return 0; if (i != pivot) { swap(A[i], A[pivot]); ans *= -1; } number tmpinv = inv(A[i][i]); for (int j = i + 1; j < n; j++) { number c = A[j][i] * tmpinv; for (int k = i; k < n; k++) { A[j][k] = (A[j][k] - c * A[i][k]); } } ans *= A[i][i]; } return ans; } // O( n^3 ). // int rank(matrix A) はまだ // O( n^3 ). // modが2の時だけ使える演算 #define FOR(x,to) for(x=0;x<(to);x++) // repに変えちゃダメ。xがint xになると動かない int gf2_rank(matrix A) { /* input */ if (!A.size() || (A.size() && A[0].size())) return 0; int n = A.size(); assert(mo == 2); int i,j,k; FOR(i,n) { int be=i,mi=n+1; for(j=i;j=n) break; FOR(j,n) swap(A[i][j],A[be][j]); FOR(j,n) if(i!=j&&A[j][mi]) { FOR(k,n) A[j][k] += A[i][k]; // ^=のつもり } } return i; } // input : a, b // output : x, y s.t. ax + by = (符号付き)gcd(a, b) int extGcd( int a, int b, int& x, int& y ) { if ( b == 0 ) { x = 1; y = 0; return a; } int g = extGcd( b, a % b, y, x ); y -= (a / b) * x; return g; } // xn = 1 (mod p) int invMod(int n, int p) { int x, y, g = extGcd ( n, p, x, y ); if (g == 1) return x; else if (g == -1) return -x; else return 0; // gcd(n, p) != 1,解なし } // 有限体上の線型方程式系 Ax = b (mod q)を解く // a = [A | b]: m × n の係数行列 // x: 解を記録するベクトル // 計算量: O(min(m, n) * m * n) bool gauss(matrix a, arr& x, int m, int n, int q) { int rank = 0; vll pivot(n); // 前進消去 for (int i = 0, j = 0; i < m && j < n-1; ++j) { int p = -1; Mod tmp = 0; // ピボットを探す for (int k = i; p < 0 && k < m; ++k) { if (a[k][j] != 0) p = k; // 有限体上なので非零で十分 } // ランク落ち対策 if (p == -1) continue; // 第i行と第p行を入れ替える for (int k = j; k < n; ++k) tmp = a[i][k], a[i][k] = a[p][k], a[p][k] = tmp; // 第i行を使って掃き出す for (int k = i+1; k < m; ++k) { tmp = - a[k][j] * invMod(a[i][j], q) % q; for (int l = j; l < n; ++l) a[k][l] += tmp * a[i][l]; } // 第i行を正規化: a[i][j] = 1 にする tmp = invMod(a[i][j], q); for (int k = j; k < n; ++k) a[i][k] = a[i][k] * tmp % q; pivot[i++] = j, rank++; } // 解の存在のチェック for (int i = rank; i < m; ++i) if (a[i][n-1] != 0) return false; // 解をxに代入(後退代入) for (int i = 0; i < rank; ++i) x[i] = a[i][n-1]; for (int i = rank-1; i >= 0; --i) { for (int j = pivot[i] + 1; j < n-1; ++j) x[i] -= a[i][j] * x[j]; // x[i] -= x[i] / q * q, x[i] = (x[i] + q) % q; // 0 <= x[i] < q に調整 } rep(i, x.size()) x[i] += Mod(0); return true; } arr solve(matrix a, arr b) { int m = a.size(); arr ret(a.size()); rep(i, a.size()) { a[i].pb(b[i]); } gauss(a, ret, m, m+1, mo); return ret; } // a x + b y = gcd(a, b)なるx, yを一つ探す。 // // g = gcd(a, b)として、 // 任意の整数kについて(x+k*b/g, y+k*a/g)が必要十分な解空間となる。 long long extgcd(long long a, long long b, long long &x, long long &y) { long long g = a; x = 1; y = 0; if (b != 0) g = extgcd(b, a % b, y, x), y -= (a / b) * x; return g; } // a x = b (mod m)を解く // これはa x + m k = bなるxの全列挙に他ならない。 // // bがgcd(a, m)で割り切れないならば、 // 解なしである。 // // 割り切れるならば // すなわち、extgcd(a, m, x, k), x *= b / gcd(a, m), k *= b / gcd(a, m)で解ける。 // 任意の整数nについて、x+n*m/gcd(a, m)が解である。 // // ret.fi, ret.seに対して、 // (1) ret.fi+k*ret.seが任意の解で、ret.fiは最小非負解 // (2) ret.fiが-1だと解なし P solveLinearEquation(ll a, ll b, ll m) { if (b % __gcd(a, m) != 0) { return P(-1, -1); } else { ll x, k; extgcd(a, m, x, k), x *= b / __gcd(a, m), k *= b / __gcd(a, m); P ret = P(x, m / __gcd(a, m)); ret.fi %= ret.se; ret.fi += ret.se; ret.fi %= ret.se; return ret; } } /*************************************/ // 謎演算 /*************************************/ // 線型連立合同式 a[i] x == b[i] (mod m[i]) (i = 0, ..., n-1) を解く. bool linearCongruences(const vector &a, const vector &b, const vector &m, long long &x, long long &M) { int n = a.size(); x = 0; M = 1; rep(i, n) { long long a_ = a[i] % M, b_ = b[i] - a[i] * x, m_ = m[i]; long long y, t, g = extgcd(a_, m_, y, t); if (b_ % g) return false; b_ /= g; m_ /= g; x += M * (y * b_ % m_); M *= m_; } x = (x + M) % M; return true; } // オイラーのφ関数 // LookUp Version const int N = 1000000; long long eulerPhi(long long n) { static int lookup = 0, p[N], f[N]; if (!lookup) { rep(i,N) p[i] = 1, f[i] = i; for (int i = 2; i < N; ++i) { if (p[i]) { f[i] -= f[i] / i; for (int j = i+i; j < N; j+=i) p[j] = 0, f[j] -= f[j] / i; } } lookup = 1; } return f[n]; } int main(void) { ll n, k; cin >> n >> k; assert(1<=n&&n<=1e16); assert(1<=k&&k<=1e5); n %= mo; constructFactorial(k+10); vector b(k+10); b[0] = 1; repi(i, 1, b.size()) { rep(j, i) { b[i] += nCr(i+1,j) * b[j]; } b[i] *= -((Mod)1 / (Mod)(i+1)); } // cout << b << endl; Mod ret = 0; rep(j, k+1) { ret += nCr(k+1, j) * b[j] * (Mod(n+1)^(k-j+1)); } ret /= Mod(k+1); cout << (ret+mo)%mo << endl; return 0; }