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: Accepted)

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

public class Main {
	static int n;
	static int[] a;
	static long[][][] dp;

	public static void main(String[] args) throws Exception {
		Scanner sc = new Scanner(System.in);
		n = sc.nextInt();
		a = new int[n];
		for (int i = 0; i < n; i++) {
			a[i] = sc.nextInt();
		}
		sc.close();

		dp = new long[2][n][n];
		for (int i = 0; i < 2; i++) {
			for (int j = 0; j < n; j++) {
				Arrays.fill(dp[i][j], Long.MIN_VALUE);
			}
		}
		System.out.println(dfs(0, 0, n - 1));
	}

	static long dfs(int t, int l, int r) {
		if (dp[t][l][r] != Long.MIN_VALUE) {
			return dp[t][l][r];
		}
		if (t == 0) {
			if (l == r) {
				return dp[t][l][r] = a[l];
			}
			long ret = Long.MIN_VALUE;
			ret = Math.max(ret, dfs(1, l + 1, r) + a[l]);
			ret = Math.max(ret, dfs(1, l, r - 1) + a[r]);
			return dp[t][l][r] = ret;
		} else {
			if (l == r) {
				return dp[t][l][r] = -a[l];
			}
			long ret = Long.MAX_VALUE;
			ret = Math.min(ret, dfs(0, l + 1, r) - a[l]);
			ret = Math.min(ret, dfs(0, l, r - 1) - a[r]);
			return dp[t][l][r] = ret;
		}
	}
}