Problem:
Score : 600 points

Problem Statement

Let us define the oddness of a permutation p = {p_1,\ p_2,\ ...,\ p_n} of {1,\ 2,\ ...,\ n} as \sum_{i = 1}^n |i - p_i|.

Find the number of permutations of {1,\ 2,\ ...,\ n} of oddness k, modulo 10^9+7.

Constraints

All values in input are integers.

1 \leq n \leq 50

0 \leq k \leq n^2

Input

Input is given from Standard Input in the following format:

n k

Output

Print the number of permutations of {1,\ 2,\ ...,\ n} of oddness k, modulo 10^9+7.

Sample Input 1

3 2

Sample Output 1

2

There are six permutations of {1,\ 2,\ 3}. Among them, two have oddness of 2: {2,\ 1,\ 3} and {1,\ 3,\ 2}.

Sample Input 2

39 14

Sample Output 2

74764168

Solution in Java8 (OpenJDK 1.8.0) (judge verdict: Accepted)

import java.io.*;
import java.util.Arrays;
import java.util.StringJoiner;
import java.util.StringTokenizer;
import java.util.function.Function;

public class Main {

    static int N, K;

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

        System.out.println(solve());
    }

    static int solve() {
        // すぬけさんのコードを写経した

        // Kは偶数でないとおかしい
        if( K%2 == 1 ) return 0;

        // あちこち二倍するのも何なので半分にする
        K = K/2;

        // i 番目まで見て
        // j 個保留して
        // k のスコア(保留してある分+確定させた分)
        int[][][] dp = new int[N+1][N+1][N*N+1];
        dp[0][0][0] = 1;

        for (int i = 0; i < N; i++) {
            for (int j = 0; j <= i; j++) {
                for (int k = 0; k <= K; k++) { // Kは増える一方なのでKより大きい値は見る必要がない
                    // 各遷移で基本的にやることは
                    // 1. つなぐ or 保留
                    // 2. 保留分のスコアが増える(距離が一つ離れるのでそうなる)
                    {
                        // ア. 今の分をつなぐ
                        int nj = j;
                        dp[i+1][nj][k+nj] += dp[i][j][k];
                        dp[i+1][nj][k+nj] %= MOD;
                    }
                    {
                        // イ. 今の分も保留する
                        int nj = j+1;
                        dp[i+1][nj][k+nj] += dp[i][j][k];
                        dp[i+1][nj][k+nj] %= MOD;
                    }
                    {
                        // ウ. 今の分と保留分を一つつなぐ
                        // 保留分 * 2(左右)のつなぎ方がある
                        int nj = j;
                        int x = j*2;
                        dp[i+1][nj][k+nj] += mul(dp[i][j][k], x);
                        dp[i+1][nj][k+nj] %= MOD;
                    }
                    {
                        // エ. 今の分と保留分を二つつなぐ
                        // 保留分^2 のつなぎ方がある
                        if( j >= 1 ) { // 保留分が残ってないとこの操作はできない
                            int nj = j-1;
                            int x = j*j;
                            dp[i+1][nj][k+nj] += mul(dp[i][j][k], x);
                            dp[i+1][nj][k+nj] %= MOD;
                        }
                    }
                }
            }

//            debug("i =", i);
//            for (int j = 0; j <= i+1; j++) {
//                for (int k = 0; k <= N*N; k++) {
//                    if( dp[i+1][j][k] != 0 ) {
//                        debug(i+1, j, k, dp[i+1][j][k]);
//                    }
//                }
//            }
        }

        // 全部つないである(保留0), スコアがK
        return dp[N][0][K];
    }

    static int MOD = 1_000_000_007;

    static int pow(int base, long exp) {
        if (exp == 0) return 1;

        int ans = 1;
        base %= MOD;
        while (exp > 0) {
            if ((exp & 1) == 1) {
                ans = mul(ans, base);
            }

            base = mul(base, base);
            exp = exp >> 1;
        }
        return ans;
    }

    static int sub(int a, int b) {
        int c = a - b;
        if (c < 0) c += MOD;
        return c;
    }

    static int div(int a, int b) {
        return mul(a, pow(b, MOD - 2));
    }

    static int add(int a, int b) {
        int c = a + b;
        if (c >= MOD) c %= MOD;
        return c;
    }

    static int mul(int a, int b) {
        long c = (long) a * b;
        if (c >= MOD) c %= MOD;
        return (int) c;
    }

    static class FermatCombination {
        private final int size;
        private final int[] factorial; // n -> factorial(n)
        private final int[] inverse;   // n -> inverse(factorial(n))

        FermatCombination(int size) {
            this.size = size;
            factorial = new int[size + 1];
            inverse = new int[size + 1];

            init();
        }

        private void init() {
            factorial[0] = 1;
            factorial[1] = 1;
            inverse[0] = 1;
            inverse[1] = 1;
            for (int i = 2; i <= size; i++) {
                factorial[i] = mul(factorial[i - 1], i);
                inverse[i] = pow(factorial[i], MOD - 2);
            }
        }

        int comb(int n, int k) {
            if (n > size) throw new RuntimeException("wtf : size=" + size + " n=" + n);
            return mul(mul(factorial[n], inverse[k]), inverse[n - k]);
        }

        // 重複組み合わせ
        // k種類からn個選ぶ場合の数
        int hcomb(int k, int n) {
            return comb(k + n - 1, n);
        }

        int group(int n, int g, int k) {
            // C(n, g) * C(n-g, g)... / k!
            // n! / (n-gk)! / g! ^ k / k!
            int ret = factorial[n];             // n!
            ret = mul(ret, inverse[n - g * k]);   // 1 / (n-gk)!
            ret = mul(ret, pow(inverse[g], k)); // 1 / (g! ^ k)
            ret = mul(ret, inverse[k]);         // 1 / k!
            return ret;
        }
    }

    @SuppressWarnings("unused")
    static class FastScanner {
        private BufferedReader reader;
        private StringTokenizer tokenizer;

        FastScanner(InputStream in) {
            reader = new BufferedReader(new InputStreamReader(in));
            tokenizer = null;
        }

        String next() {
            if (tokenizer == null || !tokenizer.hasMoreTokens()) {
                try {
                    tokenizer = new StringTokenizer(reader.readLine());
                } catch (IOException e) {
                    throw new RuntimeException(e);
                }
            }
            return tokenizer.nextToken();
        }

        String nextLine() {
            if (tokenizer == null || !tokenizer.hasMoreTokens()) {
                try {
                    return reader.readLine();
                } catch (IOException e) {
                    throw new RuntimeException(e);
                }
            }
            return tokenizer.nextToken("\n");
        }

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

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

        int[] nextIntArray(int n) {
            int[] a = new int[n];
            for (int i = 0; i < n; i++)
                a[i] = nextInt();
            return a;
        }

        int[] nextIntArray(int n, int delta) {
            int[] a = new int[n];
            for (int i = 0; i < n; i++)
                a[i] = nextInt() + delta;
            return a;
        }

        long[] nextLongArray(int n) {
            long[] a = new long[n];
            for (int i = 0; i < n; i++)
                a[i] = nextLong();
            return a;
        }
    }

    static <A> void writeLines(A[] as, Function<A, String> f) {
        PrintWriter pw = new PrintWriter(System.out);
        for (A a : as) {
            pw.println(f.apply(a));
        }
        pw.flush();
    }

    static void writeLines(int[] as) {
        PrintWriter pw = new PrintWriter(System.out);
        for (int a : as) pw.println(a);
        pw.flush();
    }

    static void writeLines(long[] as) {
        PrintWriter pw = new PrintWriter(System.out);
        for (long a : as) pw.println(a);
        pw.flush();
    }

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

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

    static void debug(Object... args) {
        StringJoiner j = new StringJoiner(" ");
        for (Object arg : args) {
            if (arg instanceof int[]) j.add(Arrays.toString((int[]) arg));
            else if (arg instanceof long[]) j.add(Arrays.toString((long[]) arg));
            else if (arg instanceof double[]) j.add(Arrays.toString((double[]) arg));
            else if (arg instanceof Object[]) j.add(Arrays.toString((Object[]) arg));
            else j.add(arg.toString());
        }
        System.err.println(j.toString());
    }
}