Problem:
Score : 600 points

Problem Statement

Given are a prime number p and a sequence of p integers a_0, \ldots, a_{p-1} consisting of zeros and ones.

Find a polynomial of degree at most p-1, f(x) = b_{p-1} x^{p-1} + b_{p-2} x^{p-2} + \ldots + b_0, satisfying the following conditions:

For each i (0 \leq i \leq p-1), b_i is an integer such that 0 \leq b_i \leq p-1.

For each i (0 \leq i \leq p-1), f(i) \equiv a_i \pmod p.

Constraints

2 \leq p \leq 2999

p is a prime number.

0 \leq a_i \leq 1

Input

Input is given from Standard Input in the following format:

p
a_0 a_1 \ldots a_{p-1}

Output

Print b_0, b_1, \ldots, b_{p-1} of a polynomial f(x) satisfying the conditions, in this order, with spaces in between.

It can be proved that a solution always exists. If multiple solutions exist, any of them will be accepted.

Sample Input 1

2
1 0

Sample Output 1

1 1

f(x) = x + 1 satisfies the conditions, as follows:

f(0) = 0 + 1 = 1 \equiv 1 \pmod 2

f(1) = 1 + 1 = 2 \equiv 0 \pmod 2

Sample Input 2

3
0 0 0

Sample Output 2

0 0 0

f(x) = 0 is also valid.

Sample Input 3

5
0 1 0 1 0

Sample Output 3

0 2 0 1 3

Solution in C# (Mono 4.6.2.0) (judge verdict: Wrong Answer)

using System;
using System.Collections.Generic;
using System.Linq;
using System.Text;
using static System.Console;
using static System.Math;

namespace AtTest.ABC_137
{
    class F
    {
        static void Main(string[] args)
        {
            var sw = new System.IO.StreamWriter(OpenStandardOutput()) { AutoFlush = false };
            SetOut(sw);

            Method(args);

            Out.Flush();
        }

        static void Method(string[] args)
        {
            int p = ReadInt();
            int[] array = ReadInts();
            CaseCalculator caseCalculator = new CaseCalculator(p, p - 1);
            long[] res = new long[p];
            for(int i = 0; i < p; i++)
            {
                if (array[i] == 0) continue;

                res[p - 1]++;
                long tmp = 1;
                for(int j = 0; j < p; j++)
                {
                    res[j] += p - (caseCalculator.Combination(p - 1, j) * tmp) % p;
                    res[j] %= p;
                    tmp *= -i;
                    tmp %= p;
                }
            }

            for(int i = 0; i < p; i++)
            {
                Write(res[i]);
                if (i < p - 1) Write(" ");
            }
        }

        class CaseCalculator
        {
            long mask;
            long[] allPermutations;

            public CaseCalculator(long mask, long permutationCnt)
            {
                this.mask = mask;
                allPermutations = AllPermutations(permutationCnt);
            }

            public long Combination(long n, long m)
            {
                if (n < m) return 0;

                if (n - m < m) m = n - m;

                return Multi(allPermutations[n],
                    Reverse(
                        Multi(allPermutations[n - m], allPermutations[m]),
                        mask - 2));
            }

            public long Permutation(long n, long m)
            {
                if (n < m) return 0;

                return Multi(allPermutations[n],
                    Reverse(allPermutations[n - m], mask - 2));
            }

            long[] AllPermutations(long n)
            {
                var perms = new long[n + 1];
                perms[0] = 1;
                for (int i = 1; i <= n; i++)
                {
                    perms[i] = Multi(perms[i - 1], i);
                }
                return perms;
            }

            public long Multi(long a, long b)
            {
                return ((a % mask) * (b % mask)) % mask;
            }

            long Reverse(long a, long pow)
            {
                if (pow == 0) return 1;
                else if (pow == 1) return a % mask;
                else
                {
                    long halfMod = Reverse(a, pow / 2);
                    long nextMod = Multi(halfMod, halfMod);
                    if (pow % 2 == 0)
                    {
                        return nextMod;
                    }
                    else
                    {
                        return Multi(nextMod, a);
                    }
                }
            }
        }

        private static string Read() { return ReadLine(); }
        private static char[] ReadChars() { return Array.ConvertAll(Read().Split(), a => a[0]); }
        private static int ReadInt() { return int.Parse(Read()); }
        private static long ReadLong() { return long.Parse(Read()); }
        private static double ReadDouble() { return double.Parse(Read()); }
        private static int[] ReadInts() { return Array.ConvertAll(Read().Split(), int.Parse); }
        private static long[] ReadLongs() { return Array.ConvertAll(Read().Split(), long.Parse); }
        private static double[] ReadDoubles() { return Array.ConvertAll(Read().Split(), double.Parse); }
    }
}