Problem: You are given a rooted tree. Let's denote d(x) as depth of node x: depth of the root is 1, depth of any other node x is d(y) + 1, where y is a parent of x.

The tree has the following property: every node x with d(x) = i has exactly ai children. Maximum possible depth of a node is n, and an = 0.

We define fk as the number of unordered pairs of vertices in the tree such that the number of edges on the simple path between them is equal to k.

Calculate fk modulo 109 + 7 for every 1 ≤ k ≤ 2n - 2.

Input

The first line of input contains an integer n (2 ≤ n ≤ 5 000) — the maximum depth of a node.

The second line of input contains n - 1 integers a1, a2, ..., an - 1 (2 ≤ ai ≤ 109), where ai is the number of children of every node x such that d(x) = i. Since an = 0, it is not given in the input.

Output

Print 2n - 2 numbers. The k-th of these numbers must be equal to fk modulo 109 + 7.

Examples

Input

4
2 2 2


Output

14 19 20 20 16 16 

Input

3
2 3


Output

8 13 6 9 

Note

This the tree from the first sample: 

<image>

Solution (in JAVA):

Solution:

import java.io.*;

/**
 * Created by dalt on 2018/4/2.
 */
public class CF954H {
    public static final int MOD = (int) (1e9 + 7);
    public static BlockReader input;
    public static PrintStream output;

    public static void main(String[] args) throws FileNotFoundException {
        if (System.getProperty("ONLINE_JUDGE") == null) {
            input = new BlockReader(new FileInputStream("D:\\DataBase\\TESTCASE\\codeforces\\CF954H.in"));
            output = System.out;
        } else {
            input = new BlockReader(System.in);
            output = new PrintStream(new BufferedOutputStream(System.out), false);
        }

        solve();

        output.flush();
    }

    public static void solve() {
        int n = input.nextInteger();
        long[] a = new long[n];
        long[] aRev = new long[n];
        long[] aChoose2Child = new long[n + 1];
        long[] aChoose2ChildRev = new long[n + 1];
        long negTwo = pow(2, MOD - 2);
        for (int i = 0, until = n - 1; i < until; i++) {
            a[i] = input.nextInteger();
            aRev[i] = pow(a[i], MOD - 2);
        }

        for (int i = 0; i < n; i++) {
            aChoose2Child[i] = a[i] * (a[i] - 1) % MOD;
            aChoose2ChildRev[i] = pow( aChoose2Child[i], MOD - 2);
        }

        long[] prefixMul = new long[n];
        long[] prefixRev = new long[n];
        prefixMul[0] = 1;
        prefixRev[0] = 1;
        for (int i = 1; i < n; i++) {
            prefixMul[i] = prefixMul[i - 1] * a[i - 1] % MOD;
            prefixRev[i] = prefixRev[i - 1] * aRev[i - 1] % MOD;
        }


        int m = (n - 1) * 2;
        long[] f = new long[m + 1];
        Loop<long[][]> dpLoop = new Loop<>(new long[m + 1][3], new long[m + 1][3]);
        final int ROTATE = 0;
        final int BALANCE = 2;
        final int LINE = 1;

        {
            long[][] next = dpLoop.turn();
            next[0][BALANCE] = 1;
        }
        for (int i = n - 2; i >= 0; i--) {
            long[][] former = dpLoop.get(0);
            long[][] next = dpLoop.turn();

            long aChooseRev = aChoose2ChildRev[i + 1] * a[i + 1] % MOD * a[i + 1] % MOD * aChoose2Child[i] % MOD;
            for (int j = 2, until = ((n - 1) - i) * 2; j <= until; j++) {
                next[j][ROTATE] = (former[j - 2][ROTATE] * aChooseRev % MOD + former[j - 2][LINE] * aChoose2Child[i] % MOD) % MOD;

                int halfJ = j >> 1;
                if ((j & 1) == 0 && i + halfJ < n) {
                    long possible = prefixMul[i + halfJ] * prefixRev[i + 1] % MOD;
                    possible = possible * possible % MOD;
                    next[j][BALANCE] = possible * aChoose2Child[i]  % MOD * negTwo % MOD;
                } else {
                    next[j][BALANCE] = 0;
                }

                if (i + j < n) {
                    next[j][LINE] = prefixMul[i + j] * prefixRev[i] % MOD;
                } else {
                    next[j][LINE] = 0;
                }
            }

            next[2][BALANCE] = aChoose2Child[i] * negTwo % MOD;
            next[1][LINE] = a[i];
            next[0][BALANCE] = 1;

            for (int j = 1; j <= m; j++) {
                f[j] = (f[j] + (next[j][LINE] + next[j][ROTATE] + next[j][BALANCE]) * prefixMul[i]) % MOD;
            }
        }

        for (int i = 1; i <= m; i++) {
            output.print(f[i]);
            output.print(' ');
        }
    }

    public static long pow(long x, int n) {
        int bit = 31 - Integer.numberOfLeadingZeros(n);
        long product = 1;
        for (; bit >= 0; bit--) {
            product = product * product % MOD;
            if (((1 << bit) & n) != 0) {
                product = product * x % MOD;
            }
        }
        return product;
    }

    public static class Loop<T> {
        T[] data;
        int offset;

        public Loop(T... data) {
            this.data = data;
        }

        public T turn() {
            offset++;
            return get(0);
        }

        public T get(int index) {
            return data[(index + offset) % data.length];
        }
    }

    public static class BlockReader {
        static final int EOF = -1;
        InputStream is;
        byte[] dBuf;
        int dPos, dSize, next;
        StringBuilder builder = new StringBuilder();

        public BlockReader(InputStream is) {
            this(is, 4096);
        }

        public BlockReader(InputStream is, int bufSize) {
            this.is = is;
            dBuf = new byte[bufSize];
            next = nextByte();
        }

        public int nextByte() {
            while (dPos >= dSize) {
                if (dSize == -1) {
                    return EOF;
                }
                dPos = 0;
                try {
                    dSize = is.read(dBuf);
                } catch (Exception e) {
                }
            }
            return dBuf[dPos++];
        }

        public String nextBlock() {
            builder.setLength(0);
            skipBlank();
            while (next != EOF && !Character.isWhitespace(next)) {
                builder.append((char) next);
                next = nextByte();
            }
            return builder.toString();
        }

        public void skipBlank() {
            while (Character.isWhitespace(next)) {
                next = nextByte();
            }
        }

        public int nextInteger() {
            skipBlank();
            int ret = 0;
            boolean rev = false;
            if (next == '+' || next == '-') {
                rev = next == '-';
                next = nextByte();
            }
            while (next >= '0' && next <= '9') {
                ret = (ret << 3) + (ret << 1) + next - '0';
                next = nextByte();
            }
            return rev ? -ret : ret;
        }

        public int nextBlock(char[] data, int offset) {
            skipBlank();
            int index = offset;
            int bound = data.length;
            while (next != EOF && index < bound && !Character.isWhitespace(next)) {
                data[index++] = (char) next;
                next = nextByte();
            }
            return index - offset;
        }

        public boolean hasMore() {
            skipBlank();
            return next != EOF;
        }
    }
}