Expression Counting

YY is a clever boy. One day, he feels very boring and writes some expressions randomly. Later, he discovers that although some expressions were written in different forms, they are essentially the same, such as a+(b-c), (a+b)-c. After several attempts, he becomes interested in this and wants to know the number of distinct expressions involving n different operands.

To make the problem easier, you can assume that only operators +, -, *, / and parentheses are permitted.

The input consists of multiple test cases.

For each test case, there is one line containing only one integer n. (1<=n<=30)
End of input is indicated by a line containing a zero.

For each test case, output the number of distinct expressions involving n different operands. Since the answer is very large, you only need to output the answer modulo 1,000,000,007.

1
2
3
4
0

1
6
68
1170

There are 6 distinct expressions involving 2 operands:
a+b, a-b, b-a, a*b, a/b, b/a.

There are 68 distinct expressions involving 3 operands:
a+b+c, a+b-c, a-b+c, a-b-c, b-a+c, b-a-c, c-a-b,
a*b*c, a*b/c, a/b*c, a/b/c, b/a*c, b/a/c, c/a/b,
a+b*c, b+a*c, c+a*b,
a+b/c, a+c/b, b+a/c, b+c/a, c+a/b, c+b/a,
a-b*c, b-a*c, c-a*b,
a-b/c, a-c/b, b-a/c, b-c/a, c-a/b, c-b/a,
b*c-a, a*c-b, a*b-c,
b/c-a, c/b-a, a/c-b, c/a-b, a/b-c, b/a-c,
a*(b+c), b*(a+c), c*(a+b),
a*(b-c), a*(c-b), b*(a-c), b*(c-a), c*(a-b), c*(b-a),
a/(b+c), b/(a+c), c/(a+b),
a/(b-c), a/(c-b), b/(a-c), b/(c-a), c/(a-b), c/(b-a),
(b+c)/a, (a+c)/b, (a+b)/c,
(b-c)/a, (c-b)/a, (a-c)/b, (c-a)/b, (a-b)/c, (b-a)/c.

When n=4, note that (a-b)/(c-d) and (b-a)/(d-c) are the same, but a/b-c/d and b/a-d/c are not.

2009 Shanghai Invitation Contest Host by DHU