結果

問題 No.2936 Sum of Square of Mex
ユーザー PNJPNJ
提出日時 2024-10-13 15:36:39
言語 PyPy3
(7.3.15)
結果
AC  
実行時間 483 ms / 2,000 ms
コード長 5,983 bytes
コンパイル時間 174 ms
コンパイル使用メモリ 82,512 KB
実行使用メモリ 133,528 KB
最終ジャッジ日時 2024-10-13 15:36:50
合計ジャッジ時間 9,814 ms
ジャッジサーバーID
（参考情報） 		judge3 / judge5
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テストケース

テストケース表示
入力 結果 実行時間
実行使用メモリ
testcase_00 AC 106 ms
85,232 KB
testcase_01 AC 105 ms
86,560 KB
testcase_02 AC 181 ms
103,356 KB
testcase_03 AC 109 ms
85,648 KB
testcase_04 AC 112 ms
85,728 KB
testcase_05 AC 108 ms
85,836 KB
testcase_06 AC 109 ms
86,056 KB
testcase_07 AC 106 ms
87,128 KB
testcase_08 AC 446 ms
129,244 KB
testcase_09 AC 292 ms
113,352 KB
testcase_10 AC 300 ms
116,072 KB
testcase_11 AC 339 ms
113,976 KB
testcase_12 AC 296 ms
116,732 KB
testcase_13 AC 241 ms
106,704 KB
testcase_14 AC 162 ms
101,028 KB
testcase_15 AC 415 ms
127,736 KB
testcase_16 AC 448 ms
133,192 KB
testcase_17 AC 439 ms
132,996 KB
testcase_18 AC 476 ms
133,528 KB
testcase_19 AC 438 ms
132,944 KB
testcase_20 AC 450 ms
133,080 KB
testcase_21 AC 466 ms
132,976 KB
testcase_22 AC 444 ms
133,136 KB
testcase_23 AC 460 ms
133,240 KB
testcase_24 AC 109 ms
86,024 KB
testcase_25 AC 112 ms
86,684 KB
testcase_26 AC 447 ms
132,900 KB
testcase_27 AC 483 ms
133,244 KB
testcase_28 AC 459 ms
133,024 KB
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ソースコード

diff # 

mod = 998244353
n = 10**6
inv = [1 for j in range(n+1)]
for a in range(2,n+1):
  # ax + py = 1 <=> rx + p(-x-qy) = -q => x = -(inv[r]) * (p//a)  (r = p % a)
  res = (mod - inv[mod%a]) * (mod // a)
  inv[a] = res % mod

fact = [1 for i in range(n+1)]
for i in range(1,n+1):
  fact[i] = fact[i-1]*i % mod

fact_inv = [1 for i in range(n+1)]
fact_inv[-1] = pow(fact[-1],mod-2,mod)
for i in range(n,0,-1):
  fact_inv[i-1] = fact_inv[i]*i % mod

def binom(n,r):
  if n < r or n < 0 or r < 0:
    return 0
  res = fact_inv[n-r] * fact_inv[r] % mod
  res *= fact[n]
  res %= mod
  return res

NTT_friend = [120586241,167772161,469762049,754974721,880803841,924844033,943718401,998244353,1045430273,1051721729,1053818881]
NTT_dict = {}
for i in range(len(NTT_friend)):
  NTT_dict[NTT_friend[i]] = i
NTT_info = [[20,74066978],[25,17],[26,30],[24,362],[23,211],[21,44009197],[22,663003469],[23,31],[20,363],[20,330],[20,2789]]

def popcount(n):
  c=(n&0x5555555555555555)+((n>>1)&0x5555555555555555)
  c=(c&0x3333333333333333)+((c>>2)&0x3333333333333333)
  c=(c&0x0f0f0f0f0f0f0f0f)+((c>>4)&0x0f0f0f0f0f0f0f0f)
  c=(c&0x00ff00ff00ff00ff)+((c>>8)&0x00ff00ff00ff00ff)
  c=(c&0x0000ffff0000ffff)+((c>>16)&0x0000ffff0000ffff)
  c=(c&0x00000000ffffffff)+((c>>32)&0x00000000ffffffff)
  return c

def topbit(n):
  h = n.bit_length()
  h -= 1
  return h

def prepared_fft(mod = 998244353):
  rank2 = NTT_info[NTT_dict[mod]][0]
  root,iroot = [0] * 30,[0] * 30
  rate2,irate2= [0] * 30,[0] * 30
  rate3,irate3= [0] * 30,[0] * 30

  root[rank2] = NTT_info[NTT_dict[mod]][1]
  iroot[rank2] = pow(root[rank2],mod - 2,mod)
  for i in range(rank2-1,-1,-1):
    root[i] = root[i+1] * root[i+1] % mod
    iroot[i] = iroot[i+1] * iroot[i+1] % mod

  prod,iprod = 1,1
  for i in range(rank2-1):
    rate2[i] = root[i + 2] * prod % mod
    irate2[i] = iroot[i + 2] * iprod % mod
    prod = prod * iroot[i + 2] % mod
    iprod = iprod * root[i + 2] % mod
  
  prod,iprod = 1,1
  for i in range(rank2-2):
    rate3[i] = root[i + 3] * prod % mod
    irate3[i] = iroot[i + 3] * iprod % mod
    prod = prod * iroot[i + 3] % mod
    iprod = iprod * root[i + 3] % mod
  
  return root,iroot,rate2,irate2,rate3,irate3

root,iroot,rate2,irate2,rate3,irate3 = prepared_fft()

def ntt(a):
  n = len(a)
  h = topbit(n)
  assert (n == 1 << h)
  le = 0
  while le < h:
    if h - le == 1:
      p = 1 << (h - le - 1)
      rot = 1
      for s in range(1 << le):
        offset = s << (h - le)
        for i in range(p):
          l = a[i + offset]
          r = a[i + offset + p] * rot % mod
          a[i + offset] = (l + r) % mod
          a[i + offset + p] = (l - r) % mod
        rot = rot * rate2[topbit(~s & -~s)] % mod
      le += 1
    else:
      p = 1 << (h - le - 2)
      rot,imag = 1,root[2]
      for s in range(1 << le):
        rot2 = rot * rot % mod
        rot3 = rot2 * rot % mod
        offset = s << (h - le)
        for i in range(p):
          a0 = a[i + offset]
          a1 = a[i + offset + p] * rot
          a2 = a[i + offset + p * 2] * rot2
          a3 = a[i + offset + p * 3] * rot3
          a1na3imag = (a1 - a3) % mod * imag
          a[i + offset] = (a0 + a2 + a1 + a3) % mod
          a[i + offset + p] = (a0 + a2 - a1 - a3) % mod
          a[i + offset + p * 2] = (a0 - a2 + a1na3imag) % mod
          a[i + offset + p * 3] = (a0 - a2 - a1na3imag) % mod
        rot = rot * rate3[topbit(~s & -~s)] % mod
      le += 2

def intt(a):
  n = len(a)
  h = topbit(n)
  assert (n == 1 << h)
  coef = pow(n,mod - 2,mod)
  for i in range(n):
    a[i] = a[i] * coef % mod
  le = h
  while le:
    if le == 1:
      p = 1 << (h - le)
      irot = 1
      for s in range(1 << (le - 1)):
        offset = s << (h - le + 1)
        for i in range(p):
          l = a[i + offset]
          r = a[i + offset + p]
          a[i + offset] = (l + r) % mod
          a[i + offset + p] = (l - r) * irot % mod
        irot = irot * irate2[topbit(~s & -~s)] % mod
      le -= 1
    else:
      p = 1 << (h - le)
      irot,iimag = 1,iroot[2]
      for s in range(1 << (le - 2)):
        irot2 = irot * irot % mod
        irot3 = irot2 * irot % mod
        offset = s << (h - le + 2)
        for i in range(p):
          a0 = a[i + offset]
          a1 = a[i + offset + p]
          a2 = a[i + offset + p * 2]
          a3 = a[i + offset + p * 3]
          a2na3iimag = (a2 - a3) * iimag % mod
          a[i + offset] = (a0 + a1 + a2 + a3) % mod
          a[i + offset + p] = (a0 - a1 + a2na3iimag) * irot % mod
          a[i + offset + p * 2] = (a0 + a1 - a2 - a3) * irot2 % mod
          a[i + offset + p * 3] = (a0 - a1 - a2na3iimag) * irot3 % mod
        irot *= irate3[topbit(~s & -~s)]
        irot %= mod
      le -= 2

def convolute_naive(a,b):
  res = [0] * (len(a) + len(b) - 1)
  for i in range(len(a)):
    for j in range(len(b)):
      res[i+j] = (res[i+j] + a[i] * b[j] % mod) % mod
  return res

def convolute(a,b):
  s = a[:]
  t = b[:]
  n = len(s)
  m = len(t)
  if min(n,m) <= 60:
    return convolute_naive(s,t)
  le = 1
  while le < n + m - 1:
    le *= 2
  s += [0] * (le - n)
  t += [0] * (le - m)
  ntt(s)
  ntt(t)
  for i in range(le):
    s[i] = s[i] * t[i] % mod
  intt(s)
  s = s[:n + m - 1]
  return s

def Taylor_Shift(f,c):
  n = len(f) - 1
  P = [f[i] * fact[i] % mod for i in range(n + 1)]
  Q = [0] * (n + 1)
  for i in range(n+1):
    Q[n-i] = pow(c,i,mod) * fact_inv[i] % mod
  g = convolute(P,Q)[n:]
  for i in range(n+1):
    g[i] *= fact_inv[i]
    g[i] %= mod
  return g

N,M = map(int,input().split())
L = min(N,M)

f = []
for i in range(L + 1):
  c = i * i % mod
  if i % 2:
    c = mod - c
  f.append(c)

f = Taylor_Shift(f,mod - 1)

ans = 0
for i in range(L + 1):
  res = pow(M - i,N,mod) * f[i] % mod
  ans = (ans + res) % mod
if M < N:
  res = 0
  for i in range(M + 2):
    c = binom(M + 1,i)
    if (M + 1 - i) % 2:
      c = mod - c
    c = c * pow(i,N,mod) % mod
    res = (res + c) % mod
  res = res * (M + 1) * (M + 1) % mod
  ans = (ans + res) % mod

print(ans)
0