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: Accepted)

import java.util.Arrays;
import java.util.Scanner;

class Main{
	static int N;
	static int[] a;

	static long[][][] memo;
	static boolean[][][] flag;

	public static void main(String[] args) {
		Scanner sc = new Scanner(System.in);
		N = sc.nextInt();

		a = new int[N];
		memo = new long[2][N][N];
		flag = new boolean[2][N][N];
		for(int i = 0; i < N; i++) {
			a[i] = sc.nextInt();
			Arrays.fill(flag[0][i], false);
			Arrays.fill(flag[1][i], false);
		}

		System.out.println(dp(0, 0, 0));
		sc.close();
	}

	static long dp(int i, int x, int y) {
		if(x + y == N) {
			return 0;
		}
		if(flag[i][x][y]) {
			return memo[i][x][y];
		}
		flag[i][x][y] = true;

		long r;
		if(i == 0) {
			memo[i][x][y] = dp(1, x + 1, y) + a[x];
			r = dp(1, x, y + 1) + a[N - y - 1];
			if(memo[i][x][y] < r)
				memo[i][x][y] = r;
		}
		else {
			memo[i][x][y] = dp(0, x + 1, y) - a[x];
			r = dp(0, x, y + 1) - a[N - y - 1];
			if(memo[i][x][y] > r)
				memo[i][x][y] = r;
		}

		return memo[i][x][y];
	}
}