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 Java (OpenJDK 11.0.6) (judge verdict: Runtime Error)

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

public class Main {
 	
  
  public static void main(String abc[]) {
     Main ans = new Main();
     System.out.println(ans.solve());
  }
  
  private long solve() {
     Scanner sc = new Scanner(System.in);
     int N = sc.nextInt();
     long [] array = new long [N+1];
     int i=0;
     long sum=0;
     while(i<=N){
        array[i++] = sc.nextLong();
        sum+= array[i-1];
     }
    
     long [][][] dp =new long[3001][3001][2];
    
     long X =  solve(dp, array, 1, 0, N);
     long Y = sum-X;
     return X-Y;
  }
  
  private long solve(long[][][] dp, long[] array, int turn, int i, int j) {
     if(i>j) return 0;
     if (dp[i][j][turn] !=0) return dp[i][j][turn];
     if (turn==1) {
         long left = array[i] + solve(dp, array, 0, i+1, j);
         long right = array[j] + solve(dp, array, 0, i, j-1);
         dp[i][j][turn] = Math.max(left,right);
     } else {
         long left = solve(dp, array, 1, i+1, j);
         long right = solve(dp, array, 1, i, j-1);
         dp[i][j][turn] = Math.min(left, right);
     }
     return dp[i][j][turn];
  }
}