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 Java8 (OpenJDK 1.8.0) (judge verdict: Time Limit Exceeded)

import java.io.*;
import java.util.*;

class Pair {
    long first, second;

    public Pair(long first, long second){
        this.first = first;
        this.second = second;
    }
}

class Main {

    // Generated by 1.1.7.1 https://github.com/kyuridenamida/atcoder-tools  (tips: You use the default template now. You can remove this line by using your custom template)
    public static void main(String[] args) throws Exception {
        final Scanner sc = new Scanner(System.in);
        long N;
        N = sc.nextLong();
        long[] a = new long[(int)(N)];
        for(int i = 0 ; i < N ; i++){
                a[i] = sc.nextLong();
        }
        solve(N, a);
    }


    static void solve(long N, long[] a){
        Pair[][] dp = new Pair[(int)N][(int)N];

        for (int i = 0; i < N; i++){
            for (int j = 0; j < N; j++){
                dp[i][j] = new Pair(0, 0);
            }
        }

        for (int i = 0; i < N; i++){
            dp[i][i].first = a[i];
        }

        for (int col = 1; col < N; col++){
            for (int i = 0; i < N - col; i++){
                int j = i + col;
                long right = dp[i][j-1].second + a[j];
                long left = dp[i+1][j].second + a[i];
                if (left > right){
                    dp[i][j].first = left;
                    dp[i][j].second = dp[i+1][j].first;
                }else{
                    dp[i][j].first = right;
                    dp[i][j].second = dp[i][j-1].first;
                }
            }
        }

        long rst = dp[0][(int)(N-1)].first - dp[0][(int)(N-1)].second;
        System.out.print(rst);


    }
}