""" Mod はグローバル変数からの指定とする. """ """ 積 """ def product_modulo(*X): y=1 for x in X: y=(x*y)%Mod return y """ 階乗 """ def Factor(N): """ 0!, 1!, ..., N! (mod Mod) を出力する. N: int """ f=[1]*(N+1) for k in range(1,N+1): f[k]=(k*f[k-1])%Mod return f def Factor_with_inverse(N): """ 0!, 1!, ..., N!, (0!)^-1, (1!)^-1, ..., (N!)^-1 を出力する. N: int """ f = Factor(N) g = [0]*(N+1) N = min(N, Mod-1) g[N] = pow(f[N], Mod - 2, Mod) for k in range(N-1,-1,-1): g[k] = ((k+1) * g[k+1]) % Mod return f, g def Double_Factor(N): """ 0!!, 1!!, ..., N!! (mod Mod) を出力する. N: int """ f=[1]*(N+1) for i in range(2,N+1): f[i]=i*f[i-2]%Mod return f def Modular_Inverse(N): """ 1^(-1), 2^(-1), ..., N^(-1) (mod Mod) を出力する. [Input] N:int [Output] [-1, 1^(-1), 2^(-1), ..., N^(-1)] (第 0 要素に注意!!) """ inv=[1]*(N+1); inv[0]=-1 for k in range(2, N+1): q,r=divmod(Mod,k) inv[k]=(-q*inv[r])%Mod return inv """ 組み合わせの数 Factor_with_inverse で fact, fact_inv を既に求めていることが前提 (グローバル変数) """ def nCr(n,r): """ nCr (1,2,...,n から相異なる r 個の整数を選ぶ方法) を求める. n,r: int """ if 0<=r<=n: return fact[n]*(fact_inv[r]*fact_inv[n-r]%Mod)%Mod else: return 0 def nPr(n,r): """ nPr (1,2,...,n から相異なる r 個の整数を選び, 並べる方法) を求める. n,r: int """ if 0<=r<=n: return (fact[n]*fact_inv[n-r])%Mod else: return 0 def nHr(n,r): """ nHr (1,2,...,n から重複を許して r 個の整数を選ぶ方法) を求める. n,r: int ※ fact, fact_inv は第 n+r-1 項まで必要 """ if n==r==0: return 1 else: return nCr(n+r-1,r) def Multinomial_Coefficient(*K): """ K=[k_0,...,k_{r-1}] に対して, k_0, ..., k_{r-1} に対する多項係数を求める. k_i: int """ N=0 g_inv=1 for k in K: N+=k g_inv*=fact_inv[k]; g_inv%=Mod return (fact[N]*g_inv)%Mod def Binomial_Coefficient_Modulo_List(n: int): """ n を固定し, r=0,1,...,n としたときの nCr (mod Mod) のリストを出力する. n: int [出力] [nC0 , nC1 ,..., nCn] """ L=[1]*(n+1) inv=Modular_Inverse(n+1) for r in range(1, n+1): L[r]=((n+1-r)*inv[r]%Mod)*L[r-1]%Mod return L def Pascal_Triangle(N: int, mode=False): """ 0<=n<=N, 0<=r<=n の全てに対して nCr (mod M) のリストを出力する. N: int [出力] [[0C0], [1C0, 1C1], ... , [nC0, ... , nCn], ..., [NC0, ..., NCN]] """ if mode: L=[[0]*(N+1) for _ in range(N+1)] L[0][0]=1 for n in range(1,N+1): Ln=L[n]; Lnn=L[n-1] Ln[0]=1 for r in range(1,N+1): Ln[r]=(Lnn[r]+Lnn[r-1])%Mod return L else: X=[1] L=[[1]] for n in range(N): Y=[1] for k in range(1, n+1): Y.append((X[k]+X[k-1])%Mod) Y.append(1) X=Y L.append(Y) return L def Lucas_Combination(n, r): """ Lucas の定理を用いて nCr (mod Mod) を求める. """ X=1 while n or r: ni=n%Mod; ri=r%Mod n//=Mod; r//=Mod if ni int: """ Mod の原始根を求める. Returns: int: Mod の原始根 """ p = Mod if p == 2: return 1 if p == 998244353: return 3 if p == 10**9 + 7: return 5 if p == 163577857: return 23 if p == 167772161: return 3 if p == 469762049: return 3 fac=[] q=2 v=p-1 while v>=q*q: e=0 while v%q==0: e+=1 v//=q if e>0: fac.append(q) q+=1 if v>1: fac.append(v) for g in range(2, p): if pow(g, p-1, p) != 1: return None flag=True for q in fac: if pow(g, (p-1) // q, p) == 1: flag = False break if flag: return g #参考元: https://judge.yosupo.jp/submission/72676 def __build_up(self): rank2=(~(Mod-1) & ((Mod-1)-1)).bit_length() root=[0]*(rank2+1); iroot=[0]*(rank2+1) rate2=[0]*max(0, rank2-1); irate2=[0]*max(0, rank2-1) rate3=[0]*max(0, rank2-2); irate3=[0]*max(0, rank2-2) root[-1]=pow(self.primitive, (Mod-1)>>rank2, Mod) iroot[-1]=pow(root[-1], -1, Mod) for i in range(rank2)[::-1]: root[i]=root[i+1]*root[i+1]%Mod iroot[i]=iroot[i+1]*iroot[i+1]%Mod prod=iprod=1 for i in range(rank2-1): rate2[i]=root[i+2]*prod%Mod irate2[i]=iroot[i+2]*prod%Mod prod*=iroot[i+2]; prod%=Mod iprod*=root[i+2]; iprod%=Mod prod=iprod = 1 for i in range(rank2-2): rate3[i]=root[i + 3]*prod%Mod irate3[i]=iroot[i + 3]*iprod%Mod prod*=iroot[i + 3]; prod%=Mod iprod*=root[i + 3]; iprod%=Mod self.root=root; self.iroot=iroot self.rate2=rate2; self.irate2=irate2 self.rate3=rate3; self.irate3=irate3 def add(self, A: list[int] | int, B: list[int] | int) -> list[int]: """ 必要ならば末尾に元を追加して, [A[i] + B[i]] を求める. """ if type(A) == list: pass elif type(A) == int: A = [A] else: raise NotImplementedError if type(B) == list: pass elif type(B) == int: B = [B] else: raise NotImplementedError m = min(len(A), len(B)) C = [(A[i] + B[i]) %Mod for i in range(m)] C.extend(A[m:]) C.extend(B[m:]) return C def sub(self, A: list[int] | int, B: list[int] | int) -> list[int]: """ 必要ならば末尾に元を追加して, [A[i] - B[i]] を求める. """ if type(A) == list: pass elif type(A) == int: A = [A] else: raise NotImplementedError if type(B) == list: pass elif type(B) == int: B = [B] else: raise NotImplementedError m = min(len(A), len(B)) C = [(A[i] - B[i]) % Mod for i in range(m)] C.extend(A[m:]) C.extend([-b % Mod for b in B[m:]]) return C def times(self, A: list[int], k: int) -> list[int]: """ [k * A[i]] を求める. """ return [k * a % Mod for a in A] #参考元 https://judge.yosupo.jp/submission/72676 def ntt(self, A: list[int]): """ A に Mod を法とする数論変換を施す ※ Mod はグローバル変数から指定 References: https://github.com/atcoder/ac-library/blob/master/atcoder/convolution.hpp https://judge.yosupo.jp/submission/72676 """ N=len(A) H=(N-1).bit_length() l=0 I=self.root[2] rate2=self.rate2; rate3=self.rate3 while l int: """ A にある非零要素の個数を求める. Args: A (list[int]): Returns: int: 非零要素の個数 """ return len(A) - A.count(0) def is_sparse(self, A: list[int], threshold: int = 25) -> bool: """A が疎かどうかを判定する. Args: A (list[int]): threshold (int, optional): 非零要素の個数が threshold 以下ならば疎と判定する. Defaults to 25. Returns: bool: 疎? """ return self.non_zero_count(A) <= threshold def coefficients_list(self, A: list[int]) -> tuple[list[int], list[int]]: """ A にある非零要素のリストを求める. Args: A (list[int]): Returns: tuple[list[int], list[int]]: ([d[0], ..., d[k-1]], [f[0], ..., f[k-1]]) の形のリスト. j = 0, 1, ..., k - 1 に対して, a[d[j]] = f[j] であることを意味する. """ f = []; d = [] for i in range(len(A)): if A[i] == 0: continue d.append(i) f.append(A[i]) return d, f def convoluton_greedy(self, A: list[int], B: list[int]) -> list[int]: """ 畳み込み積 A * B を愚直な方法で求める. Args: A (list[int]): B (list[int]): Returns: list[int]: 畳み込み積 A * B """ if len(A) < len(B): A, B = B, A n = len(A) m = len(B) C = [0] * (n + m - 1) for i in range(n): for j in range(m): C[i + j] += A[i] * B[j] % Mod for k in range(n + m - 1): C[k] %= Mod return C def convolution(self, A: list[int], B: list[int]) -> list[int]: """ 畳み込み積 A * B を求める. Args: A (list[int]): B (list[int]): Returns: list[int]: 畳み込み積 A * B """ if (not A) or (not B): return [] N=len(A) M=len(B) L=M+N-1 if min(N,M)<=50: return self.convoluton_greedy(A, B) H=L.bit_length() K=1< list[int]: """ 自分自身との畳み込み積を求める. Args: A (list[int]): Returns: list[int]: 畳み込み積 A * A """ N=len(A) L=2*N-1 if N<=50: C=[0]*L for i in range(N): for j in range(N): C[i+j]+=A[i]*A[j] C[i+j]%=Mod return C H=L.bit_length() K=1< list[int]: """ A = (A[0], A[1], ..., A[k - 1]) に対して, この k 個の畳み込み積 A[0] * A[1] * ... * A[k - 1] を求める. Args: A (list[list[int]]): 畳み込む k 個の整数のリスト Returns: list[int]: k 個の畳み込み積 A[0] * A[1] * ... * A[k - 1] """ from collections import deque if not A: return [1] Q=deque(list(range(len(A)))) A=list(A) while len(Q)>=2: i=Q.popleft(); j=Q.popleft() A[i]=self.convolution(A[i], A[j]) A[j]=None Q.append(i) i=Q.popleft() return A[i] def inverse(self, F: list[int], length: int = None) -> list[int]: """ F * G = [1, 0, 0, ..., 0] (0 が (length - 1) 個) を満たす長さ length のリスト G を求める. Args: F (list[int]): length (int, optional): 求める G の長さ. None のときは length = len(F) とする. Defaults to None. Returns: list[int]: _description_ """ M = len(F) if length is None else length if M <= 0: return [] if self.is_sparse(F): # 愚直に漸化式を用いて求める. # 計算量: F にある係数が非零の項の個数を K, 求める最大次数を N として, O(NK) 時間 d,f=self.coefficients_list(F) G=[0]*M alpha=pow(F[0], -1, Mod) G[0]=alpha for i in range(1, M): for j in range(1, len(d)): if d[j]<=i: G[i]+=f[j]*G[i-d[j]]%Mod else: break G[i]%=Mod G[i]=(-alpha*G[i])%Mod del G[M:] else: # FFTの理論を応用して求める. # 計算量: 求めたい項の個数をNとして, O(N log N) # Reference: https://judge.yosupo.jp/submission/42413 N=len(F) r=pow(F[0], -1, Mod) m=1 G=[r] while m list[int]: assert F[-1] assert G[-1] F_deg=len(F)-1 G_deg=len(G)-1 if F_deg list[int]: while F and F[-1] == 0: F.pop() while G and G[-1] == 0: G.pop() if not F: return [] return Calc.sub(F, Calc.convolution(Calc.flood_div(F, G), G)) #================================================== class Exponent_Polynomial: def __init__(self, exponent: int, polynomial: list[int]): self.exponent = exponent % Mod self.polynomial = polynomial def __repr__(self) -> str: return f"{self.__class__.__name__}({self.exponent}, {self.polynomial})" def __add__(self, other: "Exponent_Polynomial"): assert self.exponent == other.exponent return Exponent_Polynomial(self.exponent, Calc.add(self.exponent, other.exponent)) def __mul__(self, other: "Exponent_Polynomial") -> "Exponent_Polynomial": return Exponent_Polynomial((self.exponent + other.exponent) % Mod, Calc.convolution(self.polynomial, other.polynomial)) def differential(self) -> "Exponent_Polynomial": diff_poly = [0] * len(self.polynomial) for d in range(len(self.polynomial)): diff_poly[d - 1] += d * self.polynomial[d] % Mod diff_poly[d] += self.exponent * self.polynomial[d] % Mod return Exponent_Polynomial(self.exponent, [a % Mod for a in diff_poly]) def integrate_zero_to_infinity(self) -> int: if self.exponent == 0 and all(a == 0 for a in self.polynomial): return 0 res = 0 t = pow(-self.exponent, -1, Mod) alpha = 1 for d in range(len(self.polynomial)): alpha *= t; alpha %= Mod coef = fact[d] * alpha % Mod res += coef * self.polynomial[d] % Mod return res % Mod def push(self) -> "Exponent_Polynomial": return Exponent_Polynomial(self.exponent, [0] + self.polynomial) #================================================== def solve(): from itertools import product as product_iter N = int(input()) k = [0] * N; a = [0] * N for i in range(N): k[i], a[i] = map(int, input().split()) global fact, fact_inv fact, fact_inv = Factor_with_inverse(sum(k) + N) exp_polys: list[Exponent_Polynomial] = [None] * N for i in range(N): b = pow(a[i], -1, Mod) poly = [fact_inv[t] * pow(a[i], -t, Mod) % Mod for t in range(k[i])] exp_polys[i] = Exponent_Polynomial(-b, poly) product: list[Exponent_Polynomial] = [None] * (1 << N) product[0] = Exponent_Polynomial(0, [1]) expectation = [0] * (1 << N) for S in range(1, 1 << N): x = S & (-S) i = x.bit_length() - 1 product[S] = product[S ^ x] * exp_polys[i] expectation[S] = product[S].differential().push().integrate_zero_to_infinity() % Mod E_pre = [0] * (N + 1) for W in product_iter(range(3), repeat = N): S = T = U = 0 for i in range(N): if W[i] == 0: S |= 1 << i elif W[i] == 1: T |= 1 << i elif W[i] == 2: U |= 1 << i deg = (S | T).bit_count() E_pre[deg] += pow(-1, T.bit_count(), Mod) * expectation[T | U] % Mod return [sum(E_pre[m:]) % Mod for m in range(1, N + 1)] #================================================== Mod = 998244353 Calc = Calculator() print(*solve())