Introduction

Stirling numbers are a fascinating concept in combinatorial mathematics, named after the Scottish mathematician James Stirling. They play a crucial role in various fields, including algebra, number theory, and computer science. Stirling numbers come in two distinct forms: Stirling numbers of the first kind and Stirling numbers of the second kind. See more here.

The Stirling Numbers of the First Kind (unsigned) and the Second Kind are defined by the following recursions:

------- Stirling Numbers of the First Kind (Unsigned) -------
c(n, k) = (n - 1) * c(n - 1, k) + c(n - 1, k - 1)
c(0, 0) = 1
c(n, 0) = 0  for n > 0
c(0, k) = 0  for k > 0


------- Stirling Numbers of the Second Kind -------
S(n, k) = k * S(n - 1, k) + S(n - 1, k - 1)
S(0, 0) = 1
S(n, 0) = 0  for n > 0
S(0, k) = 0  for k > 0

The Challenge.

Write a function stirling_numbers(n, k) that can compute both Unsigned Stirling Numbers of the First Kind and the Second Kind and return a tuple with two values (firstKind, secondKind).

Examples.

print(stirling_numbers(0, 0)) # Output: (1, 1)
print(stirling_numbers(0, 1)) # Output: (0, 0)
print(stirling_numbers(5, 2)) # Output: (50, 15)
print(stirling_numbers(6, 2)) # Output: (274, 31)

Note: If you want a little more challenge, try to make stirling_numbers(n, k) recursive, and do not define any more functions than stirling_numbers(n, k).