Writing n as the product of k distinct positive integers (noch nicht übersetzt)

Let W(n,k) be the number of ways in which n can be written as the product of k distinct positive integers.

For example, W(144,4) = 7. There are 7 ways in which 144 can be written as a product of 4 distinct positive integers:

• 144 = 1×2×4×18
• 144 = 1×2×8×9
• 144 = 1×2×3×24
• 144 = 1×2×6×12
• 144 = 1×3×4×12
• 144 = 1×3×6×8
• 144 = 2×3×4×6

Note that permutations of the integers themselves are not considered distinct.

Furthermore, W(100!,10) modulo 1 000 000 007 = 287549200.

Find W(10000!,30) modulo 1 000 000 007.