Problem:
Score : 100 points

Problem Statement

Taro and Jiro will play the following game against each other.

Initially, they are given a sequence a = (a_1, a_2, \ldots, a_N).
Until a becomes empty, the two players perform the following operation alternately, starting from Taro:

Remove the element at the beginning or the end of a. The player earns x points, where x is the removed element.

Let X and Y be Taro's and Jiro's total score at the end of the game, respectively.
Taro tries to maximize X - Y, while Jiro tries to minimize X - Y.

Assuming that the two players play optimally, find the resulting value of X - Y.

Constraints

All values in input are integers.

1 \leq N \leq 3000

1 \leq a_i \leq 10^9

Input

Input is given from Standard Input in the following format:

N
a_1 a_2 \ldots a_N

Output

Print the resulting value of X - Y, assuming that the two players play optimally.

Sample Input 1

4
10 80 90 30

Sample Output 1

10

The game proceeds as follows when the two players play optimally (the element being removed is written bold):

Taro: (10, 80, 90, 30) → (10, 80, 90)

Jiro: (10, 80, 90) → (10, 80)

Taro: (10, 80) → (10)

Jiro: (10) → ()

Here, X = 30 + 80 = 110 and Y = 90 + 10 = 100.

Sample Input 2

3
10 100 10

Sample Output 2

-80

The game proceeds, for example, as follows when the two players play optimally:

Taro: (10, 100, 10) → (100, 10)

Jiro: (100, 10) → (10)

Taro: (10) → ()

Here, X = 10 + 10 = 20 and Y = 100.

Sample Input 3

1
10

Sample Output 3

10

Sample Input 4

10
1000000000 1 1000000000 1 1000000000 1 1000000000 1 1000000000 1

Sample Output 4

4999999995

The answer may not fit into a 32-bit integer type.

Sample Input 5

6
4 2 9 7 1 5

Sample Output 5

2

The game proceeds, for example, as follows when the two players play optimally:

Taro: (4, 2, 9, 7, 1, 5) → (4, 2, 9, 7, 1)

Jiro: (4, 2, 9, 7, 1) → (2, 9, 7, 1)

Taro: (2, 9, 7, 1) → (2, 9, 7)

Jiro: (2, 9, 7) → (2, 9)

Taro: (2, 9) → (2)

Jiro: (2) → ()

Here, X = 5 + 1 + 9 = 15 and Y = 4 + 7 + 2 = 13.

Solution in C++ (GCC 9.2.1) (judge verdict: Accepted)

#include <bits/stdc++.h>

#define CHOOSE(a) CHOOSE2 a
#define CHOOSE2(a0, a1, a2, a3, a4, x, ...) x
#define dump_1(x1) cerr << #x1 << ": " << x1 << endl
#define dump_2(x1, x2) \
  cerr << #x1 << ": " << x1 << ", " #x2 << ": " << x2 << endl
#define dump_3(x1, x2, x3)                                                \
  cerr << #x1 << ": " << x1 << ", " #x2 << ": " << x2 << ", " #x3 << ": " \
       << x3 << endl
#define dump_4(x1, x2, x3, x4)                                            \
  cerr << #x1 << ": " << x1 << ", " #x2 << ": " << x2 << ", " #x3 << ": " \
       << x3 << ", " #x4 << ": " << x4 << endl
#define dump_5(x1, x2, x3, x4, x5)                                        \
  cerr << #x1 << ": " << x1 << ", " #x2 << ": " << x2 << ", " #x3 << ": " \
       << x3 << ", " #x4 << ": " << x4 << ", " #x5 << ": " << x5 << endl
#define dump(...) \
  CHOOSE((__VA_ARGS__, dump_5, dump_4, dump_3, dump_2, dump_1, ~))(__VA_ARGS__)
#define check(s) cerr << s << endl

#define rep(i, n) for (int i = 0; i < n; i++)
#define repr(i, n) for (int i = n; i >= 0; i--)
#define FOR(i, m, n) for (int i = m; i < n; i++)
#define all(x) (x).begin(), (x).end()
#define sz(x) ((int)(x).size())
#define unique(v) v.erase(unique(v.begin(), v.end()), v.end());
#define chmax(x, y) x = max(x, y)
#define chmin(x, y) x = min(x, y)

using namespace std;

using ll = long long;

vector<int> dx = {0, 1, 0, -1};
vector<int> dy = {1, 0, -1, 0};

const ll LINF = 2e18;
const int INF = 1e9;

ll N;
std::vector<ll> a;

vector<vector<ll>> dp;

ll f(int l, int r) {
  if (dp.at(l).at(r) != -1) return dp.at(l).at(r);
  if (l == r) return a.at(l);

  ll ret = max(a.at(l) - f(l + 1, r), a.at(r) - f(l, r - 1));
  return dp.at(l).at(r) = ret;
}

void solve() {
  dp.resize(N + 1, vector<ll>(N + 1, -1));

  cout << f(0, N - 1) << endl;
}

int main() {
  cin.tie(0);
  ios::sync_with_stdio(false);
  cout << fixed << setprecision(15);
  scanf("%lld", &N);
  a.resize(N);
  for (int i = 0; i < N; i++) {
    scanf("%lld", &a[i]);
  }
  solve();
  return 0;
}