Problem:
Score : 600 points

Problem Statement

Given is an integer N. Snuke will choose integers s_1, s_2, n_1, n_2, u_1, u_2, k_1, k_2, e_1, and e_2 so that all of the following six conditions will be satisfied:

0 \leq s_1 < s_2

0 \leq n_1 < n_2

0 \leq u_1 < u_2

0 \leq k_1 < k_2

0 \leq e_1 < e_2

s_1 + s_2 + n_1 + n_2 + u_1 + u_2 + k_1 + k_2 + e_1 + e_2 \leq N

For every possible choice (s_1,s_2,n_1,n_2,u_1,u_2,k_1,k_2,e_1,e_2), compute (s_2 − s_1)(n_2 − n_1)(u_2 − u_1)(k_2 - k_1)(e_2 - e_1), and find the sum of all computed values, modulo (10^{9} +7).

Solve this problem for each of the T test cases given.

Constraints

All values in input are integers.

1 \leq T \leq 100

1 \leq N \leq 10^{9}

Input

Input is given from Standard Input in the following format:

T
\mathrm{case}_1
\vdots
\mathrm{case}_T

Each case is given in the following format:

N

Output

Print T lines. The i-th line should contain the answer to the i-th test case.

Sample Input 1

4
4
6
10
1000000000

Sample Output 1

0
11
4598
257255556

When N=4, there is no possible choice (s_1,s_2,n_1,n_2,u_1,u_2,k_1,k_2,e_1,e_2). Thus, the answer is 0.

When N=6, there are six possible choices (s_1,s_2,n_1,n_2,u_1,u_2,k_1,k_2,e_1,e_2) as follows:

(0,1,0,1,0,1,0,1,0,1)

(0,2,0,1,0,1,0,1,0,1)

(0,1,0,2,0,1,0,1,0,1)

(0,1,0,1,0,2,0,1,0,1)

(0,1,0,1,0,1,0,2,0,1)

(0,1,0,1,0,1,0,1,0,2)

We have one choice where (s_2 − s_1)(n_2 − n_1)(u_2 − u_1)(k_2 - k_1)(e_2 - e_1) is 1 and five choices where (s_2 − s_1)(n_2 − n_1)(u_2 − u_1)(k_2 - k_1)(e_2 - e_1) is 2, so the answer is 11.

Be sure to find the sum modulo (10^{9} +7).

Solution in C# (.NET Core 3.1.201) (judge verdict: Accepted)

using System;
using System.IO;
using CompLib.Mathematics;
using CompLib.Util;

public class Program
{
    public void Solve()
    {
        Console.SetOut(new StreamWriter(Console.OpenStandardOutput()) {AutoFlush = false});
        var sc = new Scanner();
        int t = sc.NextInt();
        for (int i = 0; i < t; i++)
        {
            Q(sc);
        }

        Console.Out.Flush();
    }

    void Q(Scanner sc)
    {
        long n = sc.NextInt();
        // s = 2s_1
        // ⊿s = (s2 - s1)

        // Σs + Σ⊿s <= N
        // 値は s,⊿sを分けたあと、⊿sのそれぞれから1つ選ぶ方法
        // N-5個を5 + 2 * 5 + 1個に分ける 最初5個偶数に分ける方法
        if (n < 5)
        {
            Console.WriteLine(0);
            return;
        }

        // 今見ている個数の偶奇がi 仕切りj個目入れた
        var a = new Matrix(1, 21);
        a[0, 0] = 1;

        var b = new Matrix(21, 21);
        // 最初5個
        for (int i = 0; i < 5; i++)
        {
            // 偶数個 -> 1
            for (int j = 0; j <= i; j++)
            {
                b[2 * j, 2 * i + 1] = 1;
            }

            // 奇数個 -> 偶数個
            b[2 * i + 1, 2 * i] = 1;
        }

        for (int i = 10; i <= 20; i++)
        {
            // 最初5個偶数 -> i
            for (int j = 0; j < 5; j++)
            {
                b[j * 2, i] = 1;
            }

            for (int j = 10; j <= i; j++)
            {
                b[j, i] = 1;
            }
        }

        var c = Matrix.Pow(b, n - 4);

        var d = a * c;

        Console.WriteLine(d[0, 20]);
    }

    public static void Main(string[] args) => new Program().Solve();
}

// https://bitbucket.org/camypaper/complib
namespace CompLib.Mathematics
{
    #region ModInt

    /// <summary>
    /// [0,<see cref="Mod"/>) までの値を取るような数
    /// </summary>
    public struct ModInt
    {
        /// <summary>
        /// 剰余を取る値．
        /// </summary>
        public const long Mod = (int) 1e9 + 7;

        /// <summary>
        /// 実際の数値．
        /// </summary>
        public long num;

        /// <summary>
        /// 値が <paramref name="n"/> であるようなインスタンスを構築します．
        /// </summary>
        /// <param name="n">インスタンスが持つ値</param>
        /// <remarks>パフォーマンスの問題上，コンストラクタ内では剰余を取りません．そのため，<paramref name="n"/> ∈ [0,<see cref="Mod"/>) を満たすような <paramref name="n"/> を渡してください．このコンストラクタは O(1) で実行されます．</remarks>
        public ModInt(long n)
        {
            num = n;
        }

        /// <summary>
        /// このインスタンスの数値を文字列に変換します．
        /// </summary>
        /// <returns>[0,<see cref="Mod"/>) の範囲内の整数を 10 進表記したもの．</returns>
        public override string ToString()
        {
            return num.ToString();
        }

        public static ModInt operator +(ModInt l, ModInt r)
        {
            l.num += r.num;
            if (l.num >= Mod) l.num -= Mod;
            return l;
        }

        public static ModInt operator -(ModInt l, ModInt r)
        {
            l.num -= r.num;
            if (l.num < 0) l.num += Mod;
            return l;
        }

        public static ModInt operator *(ModInt l, ModInt r)
        {
            return new ModInt(l.num * r.num % Mod);
        }

        public static implicit operator ModInt(long n)
        {
            n %= Mod;
            if (n < 0) n += Mod;
            return new ModInt(n);
        }

        /// <summary>
        /// 与えられた 2 つの数値からべき剰余を計算します．
        /// </summary>
        /// <param name="v">べき乗の底</param>
        /// <param name="k">べき指数</param>
        /// <returns>繰り返し二乗法により O(N log N) で実行されます．</returns>
        public static ModInt Pow(ModInt v, long k)
        {
            return Pow(v.num, k);
        }

        /// <summary>
        /// 与えられた 2 つの数値からべき剰余を計算します．
        /// </summary>
        /// <param name="v">べき乗の底</param>
        /// <param name="k">べき指数</param>
        /// <returns>繰り返し二乗法により O(N log N) で実行されます．</returns>
        public static ModInt Pow(long v, long k)
        {
            long ret = 1;
            for (k %= Mod - 1; k > 0; k >>= 1, v = v * v % Mod)
                if ((k & 1) == 1)
                    ret = ret * v % Mod;
            return new ModInt(ret);
        }

        /// <summary>
        /// 与えられた数の逆元を計算します．
        /// </summary>
        /// <param name="v">逆元を取る対象となる数</param>
        /// <returns>逆元となるような値</returns>
        /// <remarks>法が素数であることを仮定して，フェルマーの小定理に従って逆元を O(log N) で計算します．</remarks>
        public static ModInt Inverse(ModInt v)
        {
            return Pow(v, Mod - 2);
        }
    }

    #endregion

    #region Binomial Coefficient

    public class BinomialCoefficient
    {
        public ModInt[] fact, ifact;

        public BinomialCoefficient(int n)
        {
            fact = new ModInt[n + 1];
            ifact = new ModInt[n + 1];
            fact[0] = 1;
            for (int i = 1; i <= n; i++)
                fact[i] = fact[i - 1] * i;
            ifact[n] = ModInt.Inverse(fact[n]);
            for (int i = n - 1; i >= 0; i--)
                ifact[i] = ifact[i + 1] * (i + 1);
            ifact[0] = ifact[1];
        }

        public ModInt this[int n, int r]
        {
            get
            {
                if (n < 0 || n >= fact.Length || r < 0 || r > n) return 0;
                return fact[n] * ifact[n - r] * ifact[r];
            }
        }

        public ModInt RepeatedCombination(int n, int k)
        {
            if (k == 0) return 1;
            return this[n + k - 1, k];
        }
    }

    #endregion
}


// https://bitbucket.org/camypaper/complib
namespace CompLib.Mathematics
{
    using System.Diagnostics;
    using N = ModInt;

    #region Matrix

    public class Matrix
    {
        int row, col;
        public N[] mat;

        /// <summary>
        /// <paramref name="r"/> 行 <paramref name="c"/> 列目の要素へのアクセスを提供します。
        /// </summary>
        /// <param name="r">行の番号</param>
        /// <param name="c">列の番号</param>
        public N this[int r, int c]
        {
            get { return mat[r * col + c]; }
            set { mat[r * col + c] = value; }
        }

        public Matrix(int r, int c)
        {
            row = r;
            col = c;
            mat = new N[row * col];
        }

        public static Matrix operator *(Matrix l, Matrix r)
        {
            Debug.Assert(l.col == r.row);
            var ret = new Matrix(l.row, r.col);
            for (int i = 0; i < l.row; i++)
            for (int k = 0; k < l.col; k++)
            for (int j = 0; j < r.col; j++)
                ret.mat[i * r.col + j] = (ret.mat[i * r.col + j] + l.mat[i * l.col + k] * r.mat[k * r.col + j]);
            return ret;
        }

        /// <summary>
        /// <paramref name="m"/>^<paramref name="n"/> を O(<paramref name="m"/>^3 log <paramref name="n"/>) で計算します。
        /// </summary>
        public static Matrix Pow(Matrix m, long n)
        {
            var ret = new Matrix(m.row, m.col);
            for (int i = 0; i < m.row; i++)
                ret.mat[i * m.col + i] = 1;
            for (; n > 0; m *= m, n >>= 1)
                if ((n & 1) == 1)
                    ret = ret * m;
            return ret;
        }

        public N[][] Items
        {
            get
            {
                var a = new N[row][];
                for (int i = 0; i < row; i++)
                {
                    a[i] = new N[col];
                    for (int j = 0; j < col; j++)
                        a[i][j] = mat[i * col + j];
                }

                return a;
            }
        }
    }

    #endregion
}

namespace CompLib.Util
{
    using System;
    using System.Linq;

    class Scanner
    {
        private string[] _line;
        private int _index;
        private const char Separator = ' ';

        public Scanner()
        {
            _line = new string[0];
            _index = 0;
        }

        public string Next()
        {
            while (_index >= _line.Length)
            {
                _line = Console.ReadLine().Split(Separator);
                _index = 0;
            }

            return _line[_index++];
        }

        public int NextInt() => int.Parse(Next());
        public long NextLong() => long.Parse(Next());
        public double NextDouble() => double.Parse(Next());
        public decimal NextDecimal() => decimal.Parse(Next());
        public char NextChar() => Next()[0];
        public char[] NextCharArray() => Next().ToCharArray();

        public string[] Array()
        {
            _line = Console.ReadLine().Split(Separator);
            _index = _line.Length;
            return _line;
        }

        public int[] IntArray() => Array().Select(int.Parse).ToArray();
        public long[] LongArray() => Array().Select(long.Parse).ToArray();
        public double[] DoubleArray() => Array().Select(double.Parse).ToArray();
        public decimal[] DecimalArray() => Array().Select(decimal.Parse).ToArray();
    }
}