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.io.BufferedReader;
import java.io.IOException;
import java.io.InputStreamReader;
import java.lang.reflect.Array;
import java.util.*;

class Main{

	public static void main(String[] args) {
		FastReader f = new FastReader();
		int n = ii(f);
		long[] arr = new long[n+1];
		for(int i=1; i<=n; ++i)
			arr[i] = il(f);
		long[][] dp = new long[n+1][n+1];
		for(int i=1; i<=n-1; ++i)
			dp[i][2] = Math.abs(arr[i] - arr[i+1]);
		for(int count=3; count<=n; ++count){
			for(int i=1; i<=n-count+1; ++i){
				dp[i][count] = max(arr[i] - dp[i+1][count-1], arr[i+count-1] - dp[i][count-1]);
			}
		}
		if(n == 1)
			printn(arr[1]);
		else
			printn(dp[1][n]);
	}

	static class FastReader {
		BufferedReader br;
		StringTokenizer st;

		public FastReader() {
			br = new BufferedReader(new
					InputStreamReader(System.in));
		}

		String next() {
			while (st == null || !st.hasMoreElements()) {
				try {
					st = new StringTokenizer(br.readLine());
				} catch (IOException e) {
					e.printStackTrace();
				}
			}
			return st.nextToken();
		}

		int nextInt() {
			return Integer.parseInt(next());
		}

		long nextLong() {
			return Long.parseLong(next());
		}

		double nextDouble() {
			return Double.parseDouble(next());
		}

		String nextLine() {
			String str = "";
			try {
				str = br.readLine();
			} catch (IOException e) {
				e.printStackTrace();
			}
			return str;
		}
	}

	public static int ii(FastReader fr) {
		return fr.nextInt();
	}

	static long il(FastReader fr) {
		return fr.nextLong();
	}

	static double id(FastReader fr) {
		return fr.nextDouble();
	}

	static void printArray(Object[] arr) {
		for (Object x : arr)
			print(x, " ");
		printn("");
	}

	static void print(Object s, String sep) {
		System.out.print(s + sep);
	}

	static void printn(Object s) {
		System.out.println(s);
	}

	static long[] inputArrL(int n, FastReader fr) {
		long[] arr = new long[n];
		for (int i = 0; i < n; i++)
			arr[i] = fr.nextLong();
		return arr;
	}

	static int[] inputArr(int n, FastReader fr) {
		int[] arr = new int[n];
		for (int i = 0; i < n; i++)
			arr[i] = fr.nextInt();
		return arr;
	}

	static int max(int... args) {
		int max = Integer.MIN_VALUE;
		for (int x : args) {
			max = Math.max(x, max);
		}
		return max;
	}

	static int min(int... args) {
		int min = Integer.MAX_VALUE;
		for (int x : args) {
			min = Math.min(x, min);
		}
		return min;
	}

	static long max(long... args) {
		long max = Long.MIN_VALUE;
		for (long x : args) {
			max = Math.max(x, max);
		}
		return max;
	}

	static long min(long... args) {
		long min = Long.MAX_VALUE;
		for (long x : args) {
			min = Math.min(x, min);
		}
		return min;
	}

	static int lenDigit(int d) {
		if (d == 0)
			return 1;
		int l = 0;
		while (d != 0) {
			l++;
			d /= 10;
		}
		return l;
	}

	static int MOD = (int)1e9 + 7;

}