description: For an array c of nonnegative integers, MEX(c) denotes the smallest nonnegative integer that doesn't appear in it. For example, MEX([0, 1, 3]) = 2, MEX([42]) = 0.
  
  You are given integers n, k, and an array [b_1, b_2, …, b_n].
  
  Find the number of arrays [a_1, a_2, …, a_n], for which the following conditions hold:
  
    * 0 ≤ a_i ≤ n for each i for each i from 1 to n.
  
    * |MEX([a_1, a_2, …, a_i]) - b_i| ≤ k for each i from 1 to n.
  
  
  
  
  As this number can be very big, output it modulo 998 244 353.
  
  Input
  
  The first line of the input contains two integers n, k (1 ≤ n ≤ 2000, 0 ≤ k ≤ 50).
  
  The second line of the input contains n integers b_1, b_2, …, b_n (-k ≤ b_i ≤ n+k) — elements of the array b.
  
  Output
  
  Output a single integer — the number of arrays which satisfy the conditions from the statement, modulo 998 244 353.
  
  Examples
  
  Input
  
  
  4 0
  0 0 0 0
  
  
  Output
  
  
  256
  
  
  Input
  
  
  4 1
  0 0 0 0
  
  
  Output
  
  
  431
  
  
  Input
  
  
  4 1
  0 0 1 1
  
  
  Output
  
  
  509
  
  
  Input
  
  
  5 2
  0 0 2 2 0
  
  
  Output
  
  
  6546
  
  
  Input
  
  
  3 2
  -2 0 4
  
  
  Output
  
  
  11
solution: #include <bits/stdc++.h>
  template <typename _Tp>
  void read(_Tp &x) {
    char ch(getchar());
    bool f(false);
    while (!isdigit(ch)) f |= ch == 45, ch = getchar();
    x = ch & 15, ch = getchar();
    while (isdigit(ch)) x = x * 10 + (ch & 15), ch = getchar();
    if (f) x = -x;
  }
  template <typename _Tp, typename... Args>
  void read(_Tp &t, Args &...args) {
    read(t);
    read(args...);
  }
  const int N = 2005, mod = 998244353;
  template <typename _Tp1, typename _Tp2>
  inline void add(_Tp1 &a, _Tp2 b) {
    a = a + b >= mod ? a + b - mod : a + b;
  }
  template <typename _Tp1, typename _Tp2>
  inline void sub(_Tp1 &a, _Tp2 b) {
    a = a - b < 0 ? a - b + mod : a - b;
  }
  long long ksm(long long a, long long b = mod - 2) {
    long long res = 1;
    while (b) {
      if (b & 1) res = res * a % mod;
      a = a * a % mod, b >>= 1;
    }
    return res;
  }
  int b[N], dp[105][N], g[105][N], f[105][N];
  long long fac[N], inv[N], ifac[N];
  void init() {
    fac[0] = fac[1] = inv[0] = inv[1] = ifac[0] = ifac[1] = 1;
    for (int i = 2; i < N; ++i)
      fac[i] = fac[i - 1] * i % mod,
      inv[i] = inv[mod % i] * (mod - mod / i) % mod,
      ifac[i] = ifac[i - 1] * inv[i] % mod;
  }
  inline long long C(int n, int m) {
    return n < m || m < 0 ? 0 : fac[n] * ifac[m] % mod * ifac[n - m] % mod;
  }
  int main() {
    init();
    int n, L;
    read(n, L);
    for (int i = 1; i <= n; ++i) read(b[i]);
    dp[L][0] = 1;
    for (int i = 0; i < n; ++i) {
      memset(g, 0, sizeof(g));
      for (int j = 0; j <= L + L; ++j) {
        int v = b[i] + j - L;
        if (b[i + 1] - L <= v && v <= b[i + 1] + L) {
          for (int k = 0; k <= i; ++k)
            add(g[v - b[i + 1] + L][k], 1LL * (v + k) * dp[j][k] % mod);
          for (int k = 0; k <= i; ++k) add(g[v - b[i + 1] + L][k + 1], dp[j][k]);
        }
      }
      memset(f, 0, sizeof(f));
      for (int j = 0; j <= L + L; ++j)
        for (int k = 0; k <= i; ++k) {
          int tmp = std::max(1, -(j + b[i] - b[i + 1]));
          if (j + b[i] - b[i + 1] + tmp <= L + L && tmp - 1 <= k)
            add(f[j + b[i] - b[i + 1] + tmp][k - tmp + 1],
                1LL * dp[j][k] * fac[k] % mod * ifac[k - tmp + 1] % mod);
        }
      for (int j = 1; j <= L + L; ++j)
        for (int k = 0; k <= i; ++k)
          add(f[j][k], 1LL * f[j - 1][k + 1] * (k + 1) % mod);
      for (int j = 0; j <= L + L; ++j)
        for (int k = 0; k <= i + 1; ++k) add(g[j][k], f[j][k]);
      memcpy(dp, g, sizeof(dp));
    }
    int ans = 0;
    for (int i = 0; i <= L + L; ++i) {
      int v = b[n] + i - L;
      if (v > n) continue;
      v = n - v;
      for (int j = 0; j <= v; ++j)
        add(ans, 1LL * dp[i][j] * fac[v] % mod * ifac[v - j] % mod);
    }
    printf("%d\n", ans);
    return 0;
  }
  