Faulhaber`s Triangle#5127
The sum of the -th powers of the first integers
can be written as a polynomial of degree m+1 in :
For example:
S(n, 1) = (1 + \ldots + n) = \frac{1}{2} \cdot n^2 + \frac{1}{2} \cdot n
S(n, 2) = (1 + \ldots + n^2) = \frac{1}{3} \cdot n^3 + \frac{1}{2} \cdot n^2 + \frac{1}{6} \cdot n
S(n, 3) = (1 + \ldots + n^3) = \frac{1}{4} \cdot n^4 + \frac{1}{2} \cdot n^3 + \frac{1}{4} \cdot n^2
S(n, 4) = (1 + \ldots + n^4) = \frac{1}{5} \cdot n^5 + \frac{1}{2} \cdot n^4 + \frac{1}{3} \cdot n^3 - \frac{1}{30} \cdot n
The coefficients F(m, k) of these formulas form Faulhaber's Triangle:
where rows start with (at the top) and columns go from to m+1
Each row of Faulhaber's Triangle can be computed from the previous row by:
a) The element in row and column (j > 1) is \frac{i}{j} \cdot (the element above left); that is: F(i, j) = \frac{i}{j} \cdot F(i-1, j-1)
b) The first element in each row F(i, 1) is chosen so the sum of the elements in the row is .
Write a program to find entries in Faulhaber's Triangle as decimal fractions in lowest terms.
The first line of input contains a single integer , (1 \le P \le 1000), which is the number of data sets that follow. Each data set should be processed identically and independently.
Each data set consists of a single line of input consisting of three space separated decimal integers. The first integer is the data set number. The second integer is row number , and the third integer is the index within the row of the entry for which you are to find F(m, k), the Faulhaber's Triangle entry (0 \le m \le 400, 1 \le k \le m+1).
For each data set there is a single line of output. It contains the data set number, followed by a single space which is then followed by either the value if it is an integer OR by the numerator of the entry, a forward slash and the denominator of the entry.
Example 1