Problem:
Score : 600 points

Problem Statement

Given is a tree T with N vertices. The i-th edge connects Vertex A_i and B_i (1 \leq A_i,B_i \leq N).

Now, each vertex is painted black with probability 1/2 and white with probability 1/2, which is chosen independently from other vertices. Then, let S be the smallest subtree (connected subgraph) of T containing all the vertices painted black. (If no vertex is painted black, S is the empty graph.)

Let the holeyness of S be the number of white vertices contained in S. Find the expected holeyness of S.

Since the answer is a rational number, we ask you to print it \bmod 10^9+7, as described in Notes.

Notes

When you print a rational number, first write it as a fraction \frac{y}{x}, where x, y are integers, and x is not divisible by 10^9 + 7
(under the constraints of the problem, such representation is always possible).

Then, you need to print the only integer z between 0 and 10^9 + 6, inclusive, that satisfies xz \equiv y \pmod{10^9 + 7}.

Constraints

2 \leq N \leq 2 \times 10^5

1 \leq A_i,B_i \leq N

The given graph is a tree.

Input

Input is given from Standard Input in the following format:

N
A_1 B_1
:
A_{N-1} B_{N-1}

Output

Print the expected holeyness of S, \bmod 10^9+7.

Sample Input 1

3
1 2
2 3

Sample Output 1

125000001

If the vertices 1, 2, 3 are painted black, white, black, respectively, the holeyness of S is 1.

Otherwise, the holeyness is 0, so the expected holeyness is 1/8.

Since 8 \times 125000001 \equiv 1 \pmod{10^9+7}, we should print 125000001.

Sample Input 2

4
1 2
2 3
3 4

Sample Output 2

375000003

The expected holeyness is 3/8.

Since 8 \times 375000003 \equiv 3 \pmod{10^9+7}, we should print 375000003.

Sample Input 3

4
1 2
1 3
1 4

Sample Output 3

250000002

The expected holeyness is 1/4.

Sample Input 4

7
4 7
3 1
2 6
5 2
7 1
2 7

Sample Output 4

570312505

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

#region いつもの
using System;
using System.Collections.Generic;
using System.Linq;
using AtCoderProject;
using AtCoderProject.Reader;
using IEnumerable = System.Collections.IEnumerable;
using IEnumerator = System.Collections.IEnumerator;
using BigInteger = System.Numerics.BigInteger;
using StringBuilder = System.Text.StringBuilder;
using static AtCoderProject.Global;
using static AtCoderProject.NumGlobal;

namespace AtCoderProject { public static class Global { public static void Swap<T>(ref T a, ref T b) { T tmp = a; a = b; b = tmp; } public static T[] NewArray<T>(int len0, T value) => new T[len0].Fill(value); public static T[] NewArray<T>(int len0, Func<T> factory) { var arr = new T[len0]; for (int i = 0; i < arr.Length; i++) arr[i] = factory(); return arr; } public static T[][] NewArray<T>(int len0, int len1, T value) where T : struct { var arr = new T[len0][]; for (int i = 0; i < arr.Length; i++) arr[i] = NewArray(len1, value); return arr; } public static T[][] NewArray<T>(int len0, int len1, Func<T> factory) { var arr = new T[len0][]; for (int i = 0; i < arr.Length; i++) arr[i] = NewArray(len1, factory); return arr; } public static T[][][] NewArray<T>(int len0, int len1, int len2, T value) where T : struct { var arr = new T[len0][][]; for (int i = 0; i < arr.Length; i++) arr[i] = NewArray(len1, len2, value); return arr; } public static T[][][] NewArray<T>(int len0, int len1, int len2, Func<T> factory) { var arr = new T[len0][][]; for (int i = 0; i < arr.Length; i++) arr[i] = NewArray(len1, len2, factory); return arr; } public static T[][][][] NewArray<T>(int len0, int len1, int len2, int len3, T value) where T : struct { var arr = new T[len0][][][]; for (int i = 0; i < arr.Length; i++) arr[i] = NewArray(len1, len2, len3, value); return arr; } public static T[][][][] NewArray<T>(int len0, int len1, int len2, int len3, Func<T> factory) { var arr = new T[len0][][][]; for (int i = 0; i < arr.Length; i++) arr[i] = NewArray(len1, len2, len3, factory); return arr; } public static string AllLines<T>(IEnumerable<T> source) => string.Join("\n", source); public static string AllJoin<T>(IEnumerable<T> source) => string.Join(" ", source); public static string AllGrid<T>(IEnumerable<IEnumerable<T>> source) => AllLines(source.Select(AllJoin)); public static bool UpdateMax(ref int r, int val) { if (r < val) { r = val; return true; } return false; } public static bool UpdateMax(ref long r, long val) { if (r < val) { r = val; return true; } return false; } public static bool UpdateMin(ref int r, int val) { if (r > val) { r = val; return true; } return false; } public static bool UpdateMin(ref long r, long val) { if (r > val) { r = val; return true; } return false; } } public static class NumGlobal { public static int Pow(int x, int y) { int res = 1; for (; y > 0; y >>= 1) { if ((y & 1) == 1) res *= x; x *= x; } return res; } public static BigInteger ParseBigInteger(string s) { /* 自前実装の方が速い */ var res = BigInteger.Zero; var splited = new string[(s.Length + 7) / 8]; for (var i = 0; i < splited.Length - 1; i++) { splited[i] = s.Substring(8 * i, 8); } splited[splited.Length - 1] = s.Substring(8 * (splited.Length - 1)); foreach (var sp in splited) { res *= (int)Math.Pow(10, sp.Length); res += int.Parse(sp); } return res; } public static int BitCount(int x) { x -= (x >> 1) & 0x55555555; x = (x & 0x33333333) + ((x >> 2) & 0x33333333); x = (x + (x >> 4)) & 0x0f0f0f0f; x += x >> 8; x += x >> 16; return x & 0x3f; } public static int BitCount(long x) { x -= (x >> 1) & 0x5555555555555555; x = (x & 0x3333333333333333) + ((x >> 2) & 0x3333333333333333); x = (x + (x >> 4)) & 0x0f0f0f0f0f0f0f0f; x += x >> 8; x += x >> 16; x += x >> 32; return (int)(x & 0x0000007f); } public static int MSB(int x) { x |= x >> 1; x |= x >> 2; x |= x >> 4; x |= x >> 8; x |= x >> 16; return BitCount(x) - 1; } public static int MSB(long x) { x |= x >> 1; x |= x >> 2; x |= x >> 4; x |= x >> 8; x |= x >> 16; x |= x >> 32; return BitCount(x) - 1; } public static int LSB(int x) { x |= x << 1; x |= x << 2; x |= x << 4; x |= x << 8; x |= x << 16; return 32 - BitCount(x); } public static int LSB(long x) { x |= x << 1; x |= x << 2; x |= x << 4; x |= x << 8; x |= x << 16; x |= x << 32; return 64 - BitCount(x); } } public static class Ext { public static long ToLong(this int i) => i; public static T[] Fill<T>(this T[] arr, T value) { for (var i = 0; i < arr.Length; i++) arr[i] = value; return arr; } public static T[] Sort<T>(this T[] arr) { Array.Sort(arr); return arr; } public static string[] Sort(this string[] arr) => Sort(arr, StringComparer.OrdinalIgnoreCase); public static T[] Sort<T, U>(this T[] arr, Func<T, U> selector) where U : IComparable<U> => Sort(arr, (a, b) => selector(a).CompareTo(selector(b))); public static T[] Sort<T>(this T[] arr, Comparison<T> comparison) { Array.Sort(arr, comparison); return arr; } public static T[] Sort<T>(this T[] arr, IComparer<T> comparer) { Array.Sort(arr, comparer); return arr; } public static T[] Reverse<T>(this T[] arr) { Array.Reverse(arr); return arr; } public static Tuple<int, T> MaxBy<T>(this T[] arr) where T : IComparable<T> { T max = arr[0]; int maxIndex = 0; for (int i = 0; i < arr.Length; i++) { if (max.CompareTo(arr[i]) < 0) { max = arr[i]; maxIndex = i; } } return Tuple.Create(maxIndex, max); } public static Tuple<TSource, TMax> MaxBy<TSource, TMax>(this IEnumerable<TSource> source, Func<TSource, TMax> maxBySelector) where TMax : IComparable<TMax> { TMax max; TSource maxByItem; var e = source.GetEnumerator(); e.MoveNext(); maxByItem = e.Current; max = maxBySelector(maxByItem); while (e.MoveNext()) { var item = e.Current; var next = maxBySelector(item); if (max.CompareTo(next) < 0) { max = next; maxByItem = item; } } return Tuple.Create(maxByItem, max); } public static Tuple<int, T> MinBy<T>(this T[] arr) where T : IComparable<T> { T min = arr[0]; int minIndex = 0; for (int i = 0; i < arr.Length; i++) { if (min.CompareTo(arr[i]) > 0) { min = arr[i]; minIndex = i; } } return Tuple.Create(minIndex, min); } public static Tuple<TSource, TMin> MinBy<TSource, TMin>(this IEnumerable<TSource> source, Func<TSource, TMin> minBySelector) where TMin : IComparable<TMin> { TMin min; TSource minByItem; var e = source.GetEnumerator(); e.MoveNext(); minByItem = e.Current; min = minBySelector(minByItem); while (e.MoveNext()) { var item = e.Current; var next = minBySelector(item); if (min.CompareTo(next) > 0) { min = next; minByItem = item; } } return Tuple.Create(minByItem, min); } public static IEnumerable<TSource> Append<TSource>(this IEnumerable<TSource> source, TSource element) { foreach (var item in source) yield return item; yield return element; } public static IEnumerable<TSource> Prepend<TSource>(this IEnumerable<TSource> source, TSource element) { yield return element; foreach (var item in source) yield return item; } public static IComparer<T> Reverse<T>(this IComparer<T> comparer) => Comparer<T>.Create((x, y) => comparer.Compare(y, x)); public static Dictionary<TKey, int> GroupCount<TSource, TKey>(this IEnumerable<TSource> source, Func<TSource, TKey> keySelector) => source.GroupBy(keySelector).ToDictionary(g => g.Key, g => g.Count()); public static Dictionary<TKey, int> GroupCount<TKey>(this IEnumerable<TKey> source) => source.GroupCount(i => i); public static TValue Get<TKey, TValue>(this Dictionary<TKey, TValue> dic, TKey key) { TValue v; dic.TryGetValue(key, out v); return v; } public static TValue GetOrInit<TKey, TValue>(this Dictionary<TKey, TValue> dic, TKey key, TValue value) { TValue v; if (dic.TryGetValue(key, out v)) return v; return dic[key] = value; } } public class ReverseComparer<T> : Comparer<T> where T : IComparable<T> { private static ReverseComparer<T> defaultComparer; public static new IComparer<T> Default => defaultComparer ?? (defaultComparer = new ReverseComparer<T>()); public override int Compare(T y, T x) => x.CompareTo(y); public override bool Equals(object obj) => obj != null && GetType() == obj.GetType(); public override int GetHashCode() => GetType().GetHashCode(); } public class ΔDebugView<T> { private IEnumerable<T> collection; public ΔDebugView(IEnumerable<T> collection) { if (collection == null) throw new ArgumentNullException(nameof(collection)); this.collection = collection; }[System.Diagnostics.DebuggerBrowsable(System.Diagnostics.DebuggerBrowsableState.RootHidden)] public T[] Items => collection.ToArray(); } }
namespace AtCoderProject.Reader { using System.IO; using System.Text; using System.Diagnostics; using System.Runtime.CompilerServices;[DebuggerStepThrough] public class ConsoleReader { const int BufSize = 1 << 12; private readonly byte[] buffer = new byte[BufSize]; private readonly Stream input; private readonly Encoding encoding; private int pos = 0; private int len = 0; public ConsoleReader(Stream input, Encoding encoding) { this.input = input; this.encoding = encoding; } public ConsoleReader(Stream input) : this(input, Encoding.UTF8) { } public ConsoleReader(string text) : this(new MemoryStream(Encoding.UTF8.GetBytes(text))) { } private void MoveNext() { if (++pos >= len) { len = input.Read(buffer, 0, buffer.Length); if (len == 0) { buffer[0] = 10; } pos = 0; } } public int Int { [MethodImpl(MethodImplOptions.AggressiveInlining)] get { int res = 0; bool neg = false; while (buffer[pos] < 48) { neg = buffer[pos] == 45; MoveNext(); } do { res = checked(res * 10 + (buffer[pos] ^ 48)); MoveNext(); } while (48 <= buffer[pos]); return neg ? -res : res; } } public int Int0 => this.Int - 1; public long Long { [MethodImpl(MethodImplOptions.AggressiveInlining)] get { long res = 0; bool neg = false; while (buffer[pos] < 48) { neg = buffer[pos] == 45; MoveNext(); } do { res = res * 10 + (buffer[pos] ^ 48); MoveNext(); } while (48 <= buffer[pos]); return neg ? -res : res; } } public long Long0 => this.Long - 1; public string String { [MethodImpl(MethodImplOptions.AggressiveInlining)] get { var sb = new List<byte>(); while (buffer[pos] <= 32) MoveNext(); do { sb.Add(buffer[pos]); MoveNext(); } while (32 < buffer[pos]); return this.encoding.GetString(sb.ToArray()); } } public string Ascii { [MethodImpl(MethodImplOptions.AggressiveInlining)] get { var sb = new StringBuilder(); while (buffer[pos] <= 32) MoveNext(); do { sb.Append((char)buffer[pos]); MoveNext(); } while (32 < buffer[pos]); return sb.ToString(); } } public string Line { [MethodImpl(MethodImplOptions.AggressiveInlining)] get { var sb = new List<byte>(); while (buffer[pos] < 32) MoveNext(); do { sb.Add(buffer[pos]); MoveNext(); } while (buffer[pos] != 10 && buffer[pos] != 13); return this.encoding.GetString(sb.ToArray()); } } public double Double => double.Parse(this.Ascii);[DebuggerStepThrough] public struct RepeatReader { ConsoleReader cr; int count; public RepeatReader(ConsoleReader cr, int count) { this.cr = cr; this.count = count; } public T[] Select<T>(Func<ConsoleReader, T> factory) { var arr = new T[count]; for (var i = 0; i < count; i++) arr[i] = factory(cr); return arr; } public string[] Line { get { var arr = new string[count]; for (var i = 0; i < count; i++) arr[i] = cr.Line; return arr; } } public string[] String { get { var arr = new string[count]; for (var i = 0; i < count; i++) arr[i] = cr.String; return arr; } } public string[] Ascii { get { var arr = new string[count]; for (var i = 0; i < count; i++) arr[i] = cr.Ascii; return arr; } } public int[] Int { get { var arr = new int[count]; for (var i = 0; i < count; i++) arr[i] = cr.Int; return arr; } } public int[] Int0 { get { var arr = new int[count]; for (var i = 0; i < count; i++) arr[i] = cr.Int0; return arr; } } public long[] Long { get { var arr = new long[count]; for (var i = 0; i < count; i++) arr[i] = cr.Long; return arr; } } public long[] Long0 { get { var arr = new long[count]; for (var i = 0; i < count; i++) arr[i] = cr.Long0; return arr; } } public double[] Double { get { var arr = new double[count]; for (var i = 0; i < count; i++) arr[i] = cr.Double; return arr; } } public static implicit operator int[](RepeatReader rr) => rr.Int; public static implicit operator long[](RepeatReader rr) => rr.Long; public static implicit operator double[](RepeatReader rr) => rr.Double; public static implicit operator string[](RepeatReader rr) => rr.Ascii; } public RepeatReader Repeat(int count) => new RepeatReader(this, count);[DebuggerStepThrough] public struct SplitReader { ConsoleReader cr; public SplitReader(ConsoleReader cr) { this.cr = cr; } public string[] String { get { while (cr.buffer[cr.pos] <= 32) cr.MoveNext(); var list = new List<string>(); do { if (cr.buffer[cr.pos] < 32) cr.MoveNext(); else list.Add(cr.String); } while (cr.buffer[cr.pos] != 10 && cr.buffer[cr.pos] != 13); return list.ToArray(); } } public string[] Ascii { get { while (cr.buffer[cr.pos] <= 32) cr.MoveNext(); var list = new List<string>(); do { if (cr.buffer[cr.pos] < 32) cr.MoveNext(); else list.Add(cr.Ascii); } while (cr.buffer[cr.pos] != 10 && cr.buffer[cr.pos] != 13); return list.ToArray(); } } public int[] Int { get { while (cr.buffer[cr.pos] <= 32) cr.MoveNext(); var list = new List<int>(); do { if (cr.buffer[cr.pos] < 32) cr.MoveNext(); else list.Add(cr.Int); } while (cr.buffer[cr.pos] != 10 && cr.buffer[cr.pos] != 13); return list.ToArray(); } } public int[] Int0 { get { while (cr.buffer[cr.pos] <= 32) cr.MoveNext(); var list = new List<int>(); do { if (cr.buffer[cr.pos] < 32) cr.MoveNext(); else list.Add(cr.Int0); } while (cr.buffer[cr.pos] != 10 && cr.buffer[cr.pos] != 13); return list.ToArray(); } } public long[] Long { get { while (cr.buffer[cr.pos] <= 32) cr.MoveNext(); var list = new List<long>(); do { if (cr.buffer[cr.pos] < 32) cr.MoveNext(); else list.Add(cr.Long); } while (cr.buffer[cr.pos] != 10 && cr.buffer[cr.pos] != 13); return list.ToArray(); } } public long[] Long0 { get { while (cr.buffer[cr.pos] <= 32) cr.MoveNext(); var list = new List<long>(); do { if (cr.buffer[cr.pos] < 32) cr.MoveNext(); else list.Add(cr.Long0); } while (cr.buffer[cr.pos] != 10 && cr.buffer[cr.pos] != 13); return list.ToArray(); } } public double[] Double { get { while (cr.buffer[cr.pos] <= 32) cr.MoveNext(); var list = new List<double>(); do { if (cr.buffer[cr.pos] < 32) cr.MoveNext(); else list.Add(cr.Double); } while (cr.buffer[cr.pos] != 10 && cr.buffer[cr.pos] != 13); return list.ToArray(); } } public static implicit operator int[](SplitReader sr) => sr.Int; public static implicit operator long[](SplitReader sr) => sr.Long; public static implicit operator double[](SplitReader sr) => sr.Double; public static implicit operator string[](SplitReader sr) => sr.Ascii; } public SplitReader Split => new SplitReader(this); public static implicit operator int(ConsoleReader cr) => cr.Int; public static implicit operator long(ConsoleReader cr) => cr.Long; public static implicit operator double(ConsoleReader cr) => cr.Double; public static implicit operator string(ConsoleReader cr) => cr.Ascii; } }
public class Program
{
    [System.Diagnostics.DebuggerBrowsable(System.Diagnostics.DebuggerBrowsableState.Never)] private ConsoleReader cr; public Program(ConsoleReader consoleReader) { this.cr = consoleReader; }
    static void Main() => Console.WriteLine(new Program(new ConsoleReader(Console.OpenStandardInput())).Result());
    #endregion
    public string Result() { var obj = Calc(); if (obj is bool) return (bool)obj ? "Yes" : "No"; if (obj is double) return ((double)obj).ToString("0.####################"); return obj.ToString(); }
    private object Calc()
    {
        N = cr;
        gb = new GraphBuilder(N, cr, N - 1, false);
        graph = gb.ToTree(0);
        size = new int[N];
        Mod sum = 0;
        var Nx = Mod.Pow(2, N - 1);
        for (int i = 0; i < N; i++)
        {
            var list = new List<int>();
            list.Add(N - Size(i));
            foreach (var child in graph[i].children)
                list.Add(Size(child));
            Mod cur = Nx - 1;
            foreach (var c in list)
                cur -= Mod.Pow(2, c) - 1;
            sum += cur;
        }
        return sum / (Nx * 2);
    }


    int Size(int v)
    {
        if (size[v] > 0) return size[v];
        foreach (var child in graph[v].children)
        {
            size[v] += Size(child);
        }
        return ++size[v];
    }

    int N;
    int[] size;
    GraphBuilder gb;
    TreeNode[] graph;
}

class GraphBuilder { private List<int>[] roots; private List<int>[] children; public GraphBuilder(int count, bool isOriented) { this.roots = new List<int>[count]; this.children = new List<int>[count]; for (var i = 0; i < count; i++) { if (isOriented) { this.roots[i] = new List<int>(); this.children[i] = new List<int>(); } else { this.roots[i] = this.children[i] = new List<int>(); } } } public GraphBuilder(int count, ConsoleReader cr, int edgeCount, bool isOriented) : this(count, isOriented) { for (var i = 0; i < edgeCount; i++) this.Add(cr.Int0, cr.Int0); } public void Add(int from, int to) { children[from].Add(to); roots[to].Add(from); } public TreeNode[] ToTree(int root) { if (this.roots[0] != this.children[0]) throw new Exception("木には無向グラフをしたほうが良い"); var res = new TreeNode[this.children.Length]; res[root] = new TreeNode(root, -1, 0, this.children[root].ToArray()); var queue = new Queue<int>(); foreach (var child in res[root].children) { res[child] = new TreeNode(child, root, 1, Array.Empty<int>()); queue.Enqueue(child); } while (queue.Count > 0) { var from = queue.Dequeue(); if (res[from].root == -1) res[from].children = this.children[from].ToArray(); else { var children = new List<int>(this.children[from].Count); foreach (var c in this.children[from]) if (c != res[from].root) children.Add(c); res[from].children = children.ToArray(); } foreach (var child in res[from].children) { res[child] = new TreeNode(child, from, res[from].depth + 1, Array.Empty<int>()); queue.Enqueue(child); } } return res; } public Node[] ToArray() => Enumerable.Zip(roots, children, (r, c) => Tuple.Create(r, c)).Select((t, i) => new Node(i, t.Item1.ToArray(), t.Item2.ToArray())).ToArray(); }
class TreeNode { public TreeNode(int i, int root, int depth, int[] children) { this.index = i; this.root = root; this.children = children; this.depth = depth; } public readonly int index; public readonly int root; public readonly int depth; public int[] children; public override string ToString() => $"children: {string.Join(",", children)}"; public override bool Equals(object obj) { if (obj is TreeNode) return this.Equals((TreeNode)obj); else return false; } public bool Equals(TreeNode other) => this.index == other.index; public override int GetHashCode() => this.index; }
class Node { public Node(int i, int[] roots, int[] children) { this.index = i; this.roots = roots; this.children = children; } public readonly int index; public readonly int[] roots; public readonly int[] children; public override string ToString() => $"children: {string.Join(",", children)}"; public override bool Equals(object obj) { if (obj is Node) return this.Equals((Node)obj); else return false; } public bool Equals(Node other) => this.index == other.index; public override int GetHashCode() => this.index; }
struct Mod : IEquatable<Mod> { public const long mod = 1000000007; public static readonly Mod invalid = new Mod(-1, false); public readonly long val; private Mod(long val, bool isValid) : this(val) { if (!isValid) this.val = val; } public Mod(long val) { this.val = val % mod; if (this.val < 0) this.val += mod; } public override bool Equals(object obj) => (obj is Mod) ? this == ((Mod)obj) : false; public bool Equals(Mod obj) => this == obj; public override int GetHashCode() => val.GetHashCode(); public override string ToString() => val.ToString(); public static implicit operator Mod(long x) => new Mod(x); public static Mod operator -(Mod x) => x; public static Mod operator +(Mod x, Mod y) => (x.val + y.val) % mod; public static Mod operator -(Mod x, Mod y) => x.val >= y.val ? (x.val - y.val) % mod : (x.val - y.val) % mod + mod; public static Mod operator *(Mod x, Mod y) => (x.val * y.val) % mod; public static Mod operator /(Mod x, Mod y) => x * y.Inverse(); public static bool operator ==(Mod x, Mod y) => x.val == y.val; public static bool operator !=(Mod x, Mod y) => x.val != y.val; public Mod Inverse() { long a = val, b = mod, u = 1, v = 0; while (b > 0) { long t = a / b; var b2 = a - t * b; a = b; b = b2; var v2 = u - t * v; u = v; v = v2; } u %= mod; if (u < 0) u += mod; return u; } public static Mod Pow(Mod x, int y) { Mod res = 1; for (; y > 0; y >>= 1) { if ((y & 1) == 1) res *= x; x *= x; } return res; } public static Factor CreateFactor(int max) => new Factor(max); public class Factor { private readonly Mod[] fac, finv; public Factor(int max) { ++max; var inv = new Mod[max]; fac = new Mod[max]; finv = new Mod[max]; fac[0] = fac[1] = 1; finv[0] = finv[1] = 1; inv[1] = 1; for (var i = 2; i < max; i++) { fac[i] = fac[i - 1] * i; inv[i] = mod - inv[mod % i].val * (mod / i) % mod; finv[i] = finv[i - 1] * inv[i]; } } /** <summary>組み合わせ関数(二項係数)</summary> */ public Mod Combine(int n, int k) { if (n < k) return 0; if (n < 0 || k < 0) return 0; return fac[n] * finv[k] * finv[n - k]; } public Mod Factorial(int n) => fac[n]; public Mod FactorialInvers(int n) => finv[n]; } }