# For the sake of speed, # this convolution is specialized to mod 998244353. _fft_mod = 998244353 _fft_imag = 911660635 _fft_iimag = 86583718 _fft_rate2 = ( 911660635, 509520358, 369330050, 332049552, 983190778, 123842337, 238493703, 975955924, 603855026, 856644456, 131300601, 842657263, 730768835, 942482514, 806263778, 151565301, 510815449, 503497456, 743006876, 741047443, 56250497, 867605899, ) _fft_irate2 = ( 86583718, 372528824, 373294451, 645684063, 112220581, 692852209, 155456985, 797128860, 90816748, 860285882, 927414960, 354738543, 109331171, 293255632, 535113200, 308540755, 121186627, 608385704, 438932459, 359477183, 824071951, 103369235, ) _fft_rate3 = ( 372528824, 337190230, 454590761, 816400692, 578227951, 180142363, 83780245, 6597683, 70046822, 623238099, 183021267, 402682409, 631680428, 344509872, 689220186, 365017329, 774342554, 729444058, 102986190, 128751033, 395565204, ) _fft_irate3 = ( 509520358, 929031873, 170256584, 839780419, 282974284, 395914482, 444904435, 72135471, 638914820, 66769500, 771127074, 985925487, 262319669, 262341272, 625870173, 768022760, 859816005, 914661783, 430819711, 272774365, 530924681, ) def _butterfly(a): n = len(a) h = (n - 1).bit_length() len_ = 0 while len_ < h: if h - len_ == 1: p = 1 << (h - len_ - 1) rot = 1 for s in range(1 << len_): offset = s << (h - len_) for i in range(p): l = a[i + offset] r = a[i + offset + p] * rot a[i + offset] = (l + r) % _fft_mod a[i + offset + p] = (l - r) % _fft_mod if s + 1 != (1 << len_): rot *= _fft_rate2[(~s & -~s).bit_length() - 1] rot %= _fft_mod len_ += 1 else: p = 1 << (h - len_ - 2) rot = 1 for s in range(1 << len_): rot2 = rot * rot % _fft_mod rot3 = rot2 * rot % _fft_mod offset = s << (h - len_) 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) % _fft_mod * _fft_imag a[i + offset] = (a0 + a2 + a1 + a3) % _fft_mod a[i + offset + p] = (a0 + a2 - a1 - a3) % _fft_mod a[i + offset + p * 2] = (a0 - a2 + a1na3imag) % _fft_mod a[i + offset + p * 3] = (a0 - a2 - a1na3imag) % _fft_mod if s + 1 != (1 << len_): rot *= _fft_rate3[(~s & -~s).bit_length() - 1] rot %= _fft_mod len_ += 2 def _butterfly_inv(a): n = len(a) h = (n - 1).bit_length() len_ = h while len_: if len_ == 1: p = 1 << (h - len_) irot = 1 for s in range(1 << (len_ - 1)): offset = s << (h - len_ + 1) for i in range(p): l = a[i + offset] r = a[i + offset + p] a[i + offset] = (l + r) % _fft_mod a[i + offset + p] = (l - r) * irot % _fft_mod if s + 1 != (1 << (len_ - 1)): irot *= _fft_irate2[(~s & -~s).bit_length() - 1] irot %= _fft_mod len_ -= 1 else: p = 1 << (h - len_) irot = 1 for s in range(1 << (len_ - 2)): irot2 = irot * irot % _fft_mod irot3 = irot2 * irot % _fft_mod offset = s << (h - len_ + 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) * _fft_iimag % _fft_mod a[i + offset] = (a0 + a1 + a2 + a3) % _fft_mod a[i + offset + p] = ( (a0 - a1 + a2na3iimag) * irot % _fft_mod ) a[i + offset + p * 2] = ( (a0 + a1 - a2 - a3) * irot2 % _fft_mod ) a[i + offset + p * 3] = ( (a0 - a1 - a2na3iimag) * irot3 % _fft_mod ) if s + 1 != (1 << (len_ - 2)): irot *= _fft_irate3[(~s & -~s).bit_length() - 1] irot %= _fft_mod len_ -= 2 def _convolution_naive(a, b): n = len(a) m = len(b) ans = [0] * (n + m - 1) if n < m: for j in range(m): for i in range(n): ans[i + j] += a[i] * b[j] ans[i + j] %= _fft_mod else: for i in range(n): for j in range(m): ans[i + j] += a[i] * b[j] ans[i + j] %= _fft_mod return ans def _convolution_fft(a, b): a = a.copy() b = b.copy() n = len(a) m = len(b) z = 1 << (n + m - 2).bit_length() a += [0] * (z - n) _butterfly(a) b += [0] * (z - m) _butterfly(b) for i in range(z): a[i] *= b[i] a[i] %= _fft_mod _butterfly_inv(a) a = a[: n + m - 1] iz = pow(z, _fft_mod - 2, _fft_mod) for i in range(n + m - 1): a[i] *= iz a[i] %= _fft_mod return a def _convolution_square(a): a = a.copy() n = len(a) z = 1 << (2 * n - 2).bit_length() a += [0] * (z - n) _butterfly(a) for i in range(z): a[i] *= a[i] a[i] %= _fft_mod _butterfly_inv(a) a = a[: 2 * n - 1] iz = pow(z, _fft_mod - 2, _fft_mod) for i in range(2 * n - 1): a[i] *= iz a[i] %= _fft_mod return a def convolution(a, b): """It calculates (+, x) convolution in mod 998244353. Given two arrays a[0], a[1], ..., a[n - 1] and b[0], b[1], ..., b[m - 1], it calculates the array c of length n + m - 1, defined by > c[i] = sum(a[j] * b[i - j] for j in range(i + 1)) % 998244353. It returns an empty list if at least one of a and b are empty. Constraints ----------- > len(a) + len(b) <= 8388609 Complexity ---------- > O(n log n), where n = len(a) + len(b). """ n = len(a) m = len(b) if n == 0 or m == 0: return [] if min(n, m) <= 60: return _convolution_naive(a, b) if a is b: return _convolution_square(a) return _convolution_fft(a, b) MOD = 998244353 table_len = 10**6 + 10 fac = [1, 1] for i in range(2, table_len): fac.append(fac[-1] * i % MOD) finv = [0] * table_len finv[-1] = pow(fac[-1], MOD - 2, MOD) for i in range(table_len - 1, 0, -1): finv[i - 1] = finv[i] * i % MOD def comb(n, k): if k < 0 or n < 0 or n - k < 0: return 0 return fac[n] * finv[k] % MOD * finv[n - k] % MOD from collections import Counter P1 = 998244353 P2 = 999630629 # N = 10**5 # As = [10000] * N N = int(input()) As = list(map(int, input().split())) answer = pow(2, N - 1, P1) * sum(As) % P1 # #{ S | P2 <= \sum_{i \in S} A_i } # = #{ S' | sum(As) - P2 >= \sum_{i \in S'} A_i } weight_ub = sum(As) - P2 + 1 if weight_ub < 0: print(answer) exit() prod = [1] for A, count in Counter(As).items(): f = [0] * (weight_ub) for i in range(0, weight_ub, A): f[i] = comb(count, i // A) prod = convolution(prod, f)[:weight_ub] answer -= P2 * (sum(prod) % P1) % P1 answer %= P1 print(answer)