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 Rust (1.15.1) (judge verdict: Accepted)

use std::io::*;
use std::str::FromStr;
use std::cmp::*;

fn func(a: &Vec<i64>, dp: &mut Vec<Vec<i64>>, f: usize, l: usize, taro: bool, inf: i64) -> i64 {
    if f == l {
        return a[f] * if taro {1} else {-1};
    }

    let mut res;
    if taro {
        res = -inf;
        if dp[f+1][l] == inf {
            let po = func(a, dp, f+1, l, false, inf);
            dp[f+1][l] = po;
        }
        res = max(res, dp[f+1][l] + a[f]);

        if dp[f][l-1] == inf {
            let po = func(a, dp, f, l-1, false, inf);
            dp[f][l-1] = po;
        }
        res = max(res, dp[f][l-1] + a[l]);
    } else {
        res = inf;
        if dp[f+1][l] == inf {
            let po = func(a, dp, f+1, l, true, inf);
            dp[f+1][l] = po;
        }
        res = min(res, dp[f+1][l] - a[f]);

        if dp[f][l-1] == inf {
            let po = func(a, dp, f, l-1, true, inf);
            dp[f][l-1] = po;
        }
        res = min(res, dp[f][l-1] - a[l]);
    }
    dp[f][l] = res;
    res
}

fn main() {
    let n: usize = read();
    let a: Vec<i64> = (0..n).map(|_| read()).collect();
    let inf: i64 = 1_i64 << 60;
    
    // dp[i][j] = 区間[i,j]におけるX-Yの最大値
    let mut dp: Vec<Vec<i64>> = vec![vec![inf; n+1];n+1];
    println!("{}", func(&a, &mut dp, 0, n-1, true, inf));
}

fn read<T: FromStr>() -> T {
    let stdin = stdin();
    let stdin = stdin.lock();
    let token: String = stdin
        .bytes()
        .map(|c| c.unwrap() as char)
        .skip_while(|c| c.is_whitespace())
        .take_while(|c| !c.is_whitespace())
        .collect();
    token.parse().ok().unwrap()
}