Count the number of permutations that have a specific number of inversions.

- 1 of them has 0 inversions
- 2 of them have 1 inversion
- 2 of them have 2 inversions
- 1 of them has 3 inversions
- 0 of them have 4 inversions
- 0 of them have 5 inversions
- etc.

Given a permutationa_{1},a_{2},a_{3},...,a_{n}of thenintegers 1, 2, 3, ...,n, an inversion is a pair (a_{i},a_{j}) where i < j anda_{i}>a_{j}. The number of inversions in a permutation gives an indication on how "unsorted" a permutation is. If we wish to analyze the average running time of a sorting algorithm, it is often useful to know how many permutations ofnobjects will have a certain number of inversions.

In this problem you are asked to compute the number of permutations ofnvalues that have exactlykinversions.

For example, ifn= 3, there are 6 permutations with the indicated inversions as follows:

Therefore, for the permutations of 3 things

123 0 inversions

132

1 inversion (3 > 2)

213

1 inversion (2 > 1)

231

2 inversions (2 > 1, 3 > 1)

312

2 inversions (3 > 1, 3 > 2)

321

3 inversions (3 > 2, 3 > 1, 2 > 1)

The input consists one or more problems. The input for each problem
is specified on a single line, giving the integer *n* (1 <= *n*
<= 18) and a non-negative integer *k* (0 <= *k* <= 200).
The end of input is specified by a line with n = k = 0.

An example input file would be

column 1 1234567890 line 1:3 0[EOL] 2:3 1[EOL] 3:3 2[EOL] 4:3 3[EOL] 5:4 2[EOL] 6:4 10[EOL] 7:13 23[EOL] 8:18 80[EOL] 9:0 0[EOL] :[EOF]

For each problem, output the number of permutations of {1, ..., *n*}with
exactly *k* inversions.

The correct output corresponding to the example input file would be

column 111111111122222222223 123456789012345678901234567890 line 1:Program 8 by team 0[EOL] 2:1[EOL] 3:2[EOL] 4:2[EOL] 5:1[EOL] 6:5[EOL] 7:0[EOL] 8:46936280[EOL] 9:184348859235088[EOL] 10:End of program 8 by team 0[EOL] :[EOF]

Even though only integer arithmetic is performed, use double precision values to represent quantities to avoid overflows.