from itertools import permutations as per ############################# ############# cnb_max=10**5 mod=998244353 ############# kai=[1]*(cnb_max+1) rkai=[1]*(cnb_max+1) for i in range(cnb_max): kai[i+1]=kai[i]*(i+1)%mod rkai[cnb_max]=pow(kai[cnb_max],mod-2,mod) for i in range(cnb_max): rkai[cnb_max-1-i]=rkai[cnb_max-i]*(cnb_max-i)%mod def cnb(x,y): if y>x: return 0 if x<0:return 0 if y<0:return 0 return (kai[x]*rkai[y]%mod)*rkai[x-y]%mod def inv(n): return kai[n-1]*rkai[n]%mod ################################## # AtCoder Libary v1.4 を python に移植したもの # https://github.com/atcoder/ac-library/blob/master/atcoder/convolution.hpp #https://judge.yosupo.jp/submission/55648 MOD = 998244353 IMAG = 911660635 IIMAG = 86583718 rate2 = (0, 911660635, 509520358, 369330050, 332049552, 983190778, 123842337, 238493703, 975955924, 603855026, 856644456, 131300601, 842657263, 730768835, 942482514, 806263778, 151565301, 510815449, 503497456, 743006876, 741047443, 56250497, 867605899, 0) irate2 = (0, 86583718, 372528824, 373294451, 645684063, 112220581, 692852209, 155456985, 797128860, 90816748, 860285882, 927414960, 354738543, 109331171, 293255632, 535113200, 308540755, 121186627, 608385704, 438932459, 359477183, 824071951, 103369235, 0) rate3 = (0, 372528824, 337190230, 454590761, 816400692, 578227951, 180142363, 83780245, 6597683, 70046822, 623238099, 183021267, 402682409, 631680428, 344509872, 689220186, 365017329, 774342554, 729444058, 102986190, 128751033, 395565204, 0) irate3 = (0, 509520358, 929031873, 170256584, 839780419, 282974284, 395914482, 444904435, 72135471, 638914820, 66769500, 771127074, 985925487, 262319669, 262341272, 625870173, 768022760, 859816005, 914661783, 430819711, 272774365, 530924681, 0) def butterfly(a): n = len(a) h = (n - 1).bit_length() 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 a[i + offset] = (l + r) % MOD a[i + offset + p] = (l - r) % MOD rot *= rate2[(~s & -~s).bit_length()] rot %= MOD le += 1 else: p = 1 << (h - le - 2) rot = 1 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 *= rate3[(~s & -~s).bit_length()] rot %= MOD le += 2 def butterfly_inv(a): n = len(a) h = (n - 1).bit_length() 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 *= irate2[(~s & -~s).bit_length()] irot %= MOD le -= 1 else: p = 1 << (h - le) irot = 1 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[(~s & -~s).bit_length()] irot %= MOD le -= 2 def multiply(s, t): n = len(s) m = len(t) if min(n, m) <= 60: a = [0] * (n + m - 1) for i in range(n): if i % 8 == 0: for j in range(m): a[i + j] += s[i] * t[j] a[i + j] %= MOD else: for j in range(m): a[i + j] += s[i] * t[j] return [x % MOD for x in a] a = s.copy() b = t.copy() z = 1 << (n + m - 2).bit_length() a += [0] * (z - n) b += [0] * (z - m) butterfly(a) butterfly(b) for i in range(z): a[i] *= b[i] a[i] %= MOD butterfly_inv(a) a = a[:n + m - 1] iz = pow(z, MOD - 2, MOD) return [v * iz % MOD for v in a] def naive(n): ans=0 for p in per(range(1,n+1)): p=list(p) ma=0 cnt=0 for i in range(n): ma=max(ma,p[i]) if ma==p[i]:cnt+=1 ans+=cnt**3 return ans%mod def sol1(n): dp=[[0]*(n+3) for i in range(n+3)] dp[0][0]=1 for i in range(1,n+1): for k in range(1,n+1): for j in range(i): dp[i][k]+=dp[j][k-1]*inv(i)%mod ans=0 for i in range(1,n+1): ans+=pow(i,3,mod)*dp[n][i]*kai[n]%mod return ans%mod from collections import deque def sol2(n): def s(i): if i==0:return [1]+[0]*(n) dq=deque([]) for j in range(1,i+1): dq.append([j,1]) while len(dq)>1: f=dq.popleft() g=dq.popleft() h=multiply(f,g) dq.append(h) f=dq[0] while len(f)<=n:f.append(0) for j in range(n+1): f[j]=f[j]*rkai[i]%mod return f f=s(n) g=s(n-1) ans=0 for i in range(n+1): ans+=((f[i]-g[i])*kai[n]%mod)*pow(i,3,mod)%mod return ans%mod def sol3(n): dp=[0,0,0,0,0] dp[0]=1 for i in range(1,n+1): for j in range(3,-1,-1): dp[j]+=inv(i)*dp[j-1] dp[j]%=mod ans=dp[1]+3*2*dp[2]+6*dp[3] return ans*kai[n]%mod n=int(input()) print(sol1(n)) print(sol2(n)) print(sol3(n)) #print(naive(n))