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 C# (Mono 4.6.2.0) (judge verdict: Accepted)

using System;
using System.Collections.Generic;
using System.Diagnostics;
using System.Globalization;
using System.IO;
using System.Linq;
using System.Numerics;
using System.Text;
using System.Text.RegularExpressions;
using System.Threading;

namespace ReadWriteTemplate
{
    public static class Solver
    {
        private static void SolveCase()
        {
            int n = ReadInt();
            long[] a = ReadLongArray();

            long[,] b = new long[n, n];
            for (int i = 0; i < n; i++)
            {
                b[i, i] = (n % 2 == 1) ? a[i] : -a[i];
            }

            for (int len = 1; len < n; len++)
            {
                for (int i = 0; i < n - len; i++)
                {
                    int j = i + len;
                    if ((n + len) % 2 == 1)
                    {
                        b[i, j] = Math.Max(b[i, j - 1] + a[j], a[i] + b[i + 1, j]);
                    }
                    else
                    {
                        b[i, j] = Math.Min(b[i, j - 1] - a[j], -a[i] + b[i + 1, j]);
                    }
                }
            }

            long ans = b[0, n - 1];

            Writer.WriteLine(ans);
        }

        public static void Solve()
        {
#if DEBUG
            var sw = Stopwatch.StartNew();
#endif

            SolveCase();

            /*int T = ReadInt();
            for (int i = 0; i < T; i++)
            {
                // Writer.WriteLine("Case {0}:", i + 1);
                // Writer.Write("Case #{0}: ", i + 1);
                SolveCase();
            }*/

#if DEBUG
            sw.Stop();
            Console.WriteLine($"{sw.ElapsedMilliseconds} ms");
#endif
        }

        public static void Main()
        {
            Thread.CurrentThread.CurrentCulture = CultureInfo.InvariantCulture;

#if DEBUG
            // Reader = Console.In; Writer = Console.Out;
            Reader = File.OpenText("input.txt"); Writer = File.CreateText("output.txt");
#else
            Reader = Console.In; Writer = Console.Out;
#endif

            // Solve();
            Thread thread = new Thread(Solve, 64 * 1024 * 1024);
            thread.CurrentCulture = CultureInfo.InvariantCulture;
            thread.Start();
            thread.Join();

            Reader.Close();
            Writer.Close();
        }

        public static IOrderedEnumerable<TSource> OrderByWithShuffle<TSource, TKey>(this IEnumerable<TSource> source, Func<TSource, TKey> keySelector)
        {
            return source.Shuffle().OrderBy(keySelector);
        }

        public static T[] Shuffle<T>(this IEnumerable<T> source)
        {
            T[] result = source.ToArray();
            Random rnd = new Random();
            for (int i = result.Length - 1; i >= 1; i--)
            {
                int k = rnd.Next(i + 1);
                T tmp = result[k];
                result[k] = result[i];
                result[i] = tmp;
            }
            return result;
        }

        #region Read/Write

        private static TextReader Reader;

        private static TextWriter Writer;

        private static Queue<string> CurrentLineTokens = new Queue<string>();

        private static string[] ReadAndSplitLine()
        {
            return Reader.ReadLine().Split(new[] { ' ', '\t' }, StringSplitOptions.RemoveEmptyEntries);
        }

        public static string ReadToken()
        {
            while (CurrentLineTokens.Count == 0)
                CurrentLineTokens = new Queue<string>(ReadAndSplitLine());
            return CurrentLineTokens.Dequeue();
        }

        public static string ReadLine()
        {
            return Reader.ReadLine();
        }

        public static int ReadInt()
        {
            return int.Parse(ReadToken());
        }

        public static long ReadLong()
        {
            return long.Parse(ReadToken());
        }

        public static double ReadDouble()
        {
            return double.Parse(ReadToken(), CultureInfo.InvariantCulture);
        }

        public static int[] ReadIntArray()
        {
            return ReadAndSplitLine().Select(int.Parse).ToArray();
        }

        public static long[] ReadLongArray()
        {
            return ReadAndSplitLine().Select(long.Parse).ToArray();
        }

        public static double[] ReadDoubleArray()
        {
            return ReadAndSplitLine().Select(s => double.Parse(s, CultureInfo.InvariantCulture)).ToArray();
        }

        public static int[][] ReadIntMatrix(int numberOfRows)
        {
            int[][] matrix = new int[numberOfRows][];
            for (int i = 0; i < numberOfRows; i++)
                matrix[i] = ReadIntArray();
            return matrix;
        }

        public static string[] ReadLines(int quantity)
        {
            string[] lines = new string[quantity];
            for (int i = 0; i < quantity; i++)
                lines[i] = Reader.ReadLine().Trim();
            return lines;
        }

        public static void WriteArray<T>(IEnumerable<T> array)
        {
            Writer.WriteLine(string.Join(" ", array));
        }

        #endregion
    }
}