#include using namespace std; using ll = long long; using uint = unsigned int; using ull = unsigned long long; #define rep(i,n) for(int i=0;i=0;i--) #define per1(i,n) for(int i=int(n);i>0;i--) #define all(c) c.begin(),c.end() #define si(x) int(x.size()) #define pb push_back #define eb emplace_back #define fs first #define sc second template using V = vector; template using VV = vector>; template bool chmax(T& x, U y){ if(x bool chmin(T& x, U y){ if(y void mkuni(V& v){sort(all(v));v.erase(unique(all(v)),v.end());} template int lwb(const V& v, const T& a){return lower_bound(all(v),a) - v.begin();} template V Vec(size_t a) { return V(a); } template auto Vec(size_t a, Ts... ts) { return V(ts...))>(a, Vec(ts...)); } template ostream& operator<<(ostream& o,const pair &p){ return o<<"("< ostream& operator<<(ostream& o,const vector &vc){ o<<"{"; for(const T& v:vc) o< D divFloor(D a, D b){ return a / b - (((a ^ b) < 0 && a % b != 0) ? 1 : 0); } template D divCeil(D a, D b) { return a / b + (((a ^ b) > 0 && a % b != 0) ? 1 : 0); } template struct ModInt{ using uint = unsigned int; using ll = long long; using ull = unsigned long long; constexpr static uint mod = mod_; uint v; ModInt():v(0){} ModInt(ll _v):v(normS(_v%mod+mod)){} explicit operator bool() const {return v!=0;} static uint normS(const uint &x){return (x [0 , mod-1] static ModInt make(const uint &x){ModInt m; m.v=x; return m;} ModInt operator+(const ModInt& b) const { return make(normS(v+b.v));} ModInt operator-(const ModInt& b) const { return make(normS(v+mod-b.v));} ModInt operator-() const { return make(normS(mod-v)); } ModInt operator*(const ModInt& b) const { return make((ull)v*b.v%mod);} ModInt operator/(const ModInt& b) const { return *this*b.inv();} ModInt& operator+=(const ModInt& b){ return *this=*this+b;} ModInt& operator-=(const ModInt& b){ return *this=*this-b;} ModInt& operator*=(const ModInt& b){ return *this=*this*b;} ModInt& operator/=(const ModInt& b){ return *this=*this/b;} ModInt& operator++(int){ return *this=*this+1;} ModInt& operator--(int){ return *this=*this-1;} template friend ModInt operator+(T a, const ModInt& b){ return (ModInt(a) += b);} template friend ModInt operator-(T a, const ModInt& b){ return (ModInt(a) -= b);} template friend ModInt operator*(T a, const ModInt& b){ return (ModInt(a) *= b);} template friend ModInt operator/(T a, const ModInt& b){ return (ModInt(a) /= b);} ModInt pow(ll p) const { if(p<0) return inv().pow(-p); ModInt a = 1; ModInt x = *this; while(p){ if(p&1) a *= x; x *= x; p >>= 1; } return a; } ModInt inv() const { // should be prime return pow(mod-2); } // ll extgcd(ll a,ll b,ll &x,ll &y) const{ // ll p[]={a,1,0},q[]={b,0,1}; // while(*q){ // ll t=*p/ *q; // rep(i,3) swap(p[i]-=t*q[i],q[i]); // } // if(p[0]<0) rep(i,3) p[i]=-p[i]; // x=p[1],y=p[2]; // return p[0]; // } // ModInt inv() const { // ll x,y; // extgcd(v,mod,x,y); // return make(normS(x+mod)); // } bool operator==(const ModInt& b) const { return v==b.v;} bool operator!=(const ModInt& b) const { return v!=b.v;} bool operator<(const ModInt& b) const { return v>(istream &o,ModInt& x){ ll tmp; o>>tmp; x=ModInt(tmp); return o; } friend ostream& operator<<(ostream &o,const ModInt& x){ return o<; //using mint = ModInt<1000000007>; V fact,ifact,invs; mint Choose(int a,int b){ if(b<0 || a=0;i--) ifact[i] = ifact[i+1] * (i+1); rep1(i,N-1) invs[i] = fact[i-1] * ifact[i]; } // inplace_fmt (without bit rearranging) // fft: // a[rev(i)] <- \sum_j \zeta^{ij} a[j] // invfft: // a[i] <- (1/n) \sum_j \zeta^{-ij} a[rev(j)] // These two are inversions. // !!! CHANGE IF MOD is unusual !!! const int ORDER_2_MOD_MINUS_1 = 23; // ord_2 (mod-1) const mint PRIMITIVE_ROOT = 3; // primitive root of (Z/pZ)* void fft(V& a){ static constexpr uint mod = mint::mod; static constexpr uint mod2 = mod + mod; static const int H = ORDER_2_MOD_MINUS_1; static const mint root = PRIMITIVE_ROOT; static mint magic[H-1]; int n = si(a); assert(!(n & (n-1))); assert(n >= 1); assert(n <= 1<>(i+2))*3); magic[i] = w; } } int m = n; if(m >>= 1){ rep(i,m){ uint v = a[i+m].v; // < M a[i+m].v = a[i].v + mod - v; // < 2M a[i].v += v; // < 2M } } if(m >>= 1){ mint p = 1; for(int h=0,s=0; s>= 1){ mint p = 1; for(int h=0,s=0; s>= 1){ mint p = 1; for(int h=0,s=0; s= mod2) ? a[i].v - mod2 : a[i].v; // < 2M a[i+m].v = a[i].v + mod - v; // < 3M a[i].v += v; // < 3M } p *= magic[__builtin_ctz(++h)]; } } } rep(i,n){ a[i].v = (a[i].v >= mod2) ? a[i].v - mod2 : a[i].v; // < 2M a[i].v = (a[i].v >= mod) ? a[i].v - mod : a[i].v; // < M } // finally < mod !! } void invfft(V& a){ static constexpr uint mod = mint::mod; static constexpr uint mod2 = mod + mod; static const int H = ORDER_2_MOD_MINUS_1; static const mint root = PRIMITIVE_ROOT; static mint magic[H-1]; int n = si(a); assert(!(n & (n-1))); assert(n >= 1); assert(n <= 1<>(i+2))*3); magic[i] = w.inv(); } } int m = 1; if(m < n>>1){ mint p = 1; for(int h=0,s=0; s>1; m <<= 1){ mint p = 1; for(int h=0,s=0; s>1);i++){ ull x = a[i].v + mod2 - a[i+m].v; // < 4M a[i].v += a[i+m].v; // < 4M a[i].v = (a[i].v >= mod2) ? a[i].v - mod2 : a[i].v; // < 2M a[i+m].v = (p.v * x) % mod; // < M } for(int i=s+(m>>1); i 70ms // verify https://judge.yosupo.jp/submission/44937 V multiply(V a, V b) { int A = si(a), B = si(b); if (!A || !B) return {}; int n = A+B-1; int s = 1; while(s 2 a.resize(s); fft(a); rep(i,s) a[i] *= a[i]; }else{ a.resize(s); fft(a); b.resize(s); fft(b); rep(i,s) a[i] *= b[i]; } invfft(a); a.resize(n); return a; } int main(){ cin.tie(0); ios::sync_with_stdio(false); //DON'T USE scanf/printf/puts !! cout << fixed << setprecision(20); InitFact(500010); int N; ll K; cin >> N >> K; N++; V A(N); rep(i,N) cin >> A[i]; VV G(N); rep1(i,N-1){ int p; cin >> p; G[p].pb(i); } V dep(N); per(i,N) for(int j: G[i]) chmax(dep[i],dep[j]+1); V ans(N); V> s(N); // first VV t(N); // last auto dfs1 = [&](auto& self,int v) -> void{ if(G[v].empty()){ s[v].push_back(A[v]); return; } for(int u: G[v]) self(self,u); int heavy = G[v][0]; for(int u: G[v]) if(dep[u] > dep[heavy]) heavy = u; swap(s[v],s[heavy]); for(int u: G[v]) if(u != heavy){ rep(i,si(s[u])) s[v][i] += s[u][i]; } s[v].push_front(A[v]); }; dfs1(dfs1,0); auto dfs = [&](auto& self,int v,deque& deq) -> void{ if(deq.empty()){ int n = si(s[v]); V f(n); rep(i,n) f[i] = s[v][i]; V g(n); { mint C = 1; rep(i,n){ g[i] = C; assert(C == Choose(K,i)); C *= K-i; C *= invs[i+1]; } show(g); reverse(all(g)); } auto h = multiply(f,g); t[v].resize(n); rep(i,n) t[v][i] = h[n-1+i]; deq.resize(n); rep(i,n) deq[i] = t[v][i]; } show(si(deq)); show(v); show(deq[0]); show("--------"); ans[v] = deq[0]; deq.pop_front(); if(G[v].empty()) return; int heavy = G[v][0]; for(int u: G[v]) if(dep[u] > dep[heavy]) heavy = u; for(int u: G[v]) if(u != heavy){ deque _; self(self,u,_); rep(i,dep[u]+1) deq[i] -= t[u][i]; } self(self,heavy,deq); }; deque _; dfs(dfs,0,_); rep(v,N) cout << ans[v] << endl; }