p

H

```
"""
Print all the Catalan numbers from 0 to n, n being the user input.
* The Catalan numbers are a sequence of positive integers that
* appear in many counting problems in combinatorics [1]. Such
* problems include counting [2]:
* - The number of Dyck words of length 2n
* - The number well-formed expressions with n pairs of parentheses
* (e.g., `()()` is valid but `())(` is not)
* - The number of different ways n + 1 factors can be completely
* parenthesized (e.g., for n = 2, C(n) = 2 and (ab)c and a(bc)
* are the two valid ways to parenthesize.
* - The number of full binary trees with n + 1 leaves
* A Catalan number satisfies the following recurrence relation
* which we will use in this algorithm [1].
* C(0) = C(1) = 1
* C(n) = sum(C(i).C(n-i-1)), from i = 0 to n-1
* In addition, the n-th Catalan number can be calculated using
* the closed form formula below [1]:
* C(n) = (1 / (n + 1)) * (2n choose n)
* Sources:
* [1] https://brilliant.org/wiki/catalan-numbers/
* [2] https://en.wikipedia.org/wiki/Catalan_number
"""
def catalan_numbers(upper_limit: int) -> "list[int]":
"""
Return a list of the Catalan number sequence from 0 through `upper_limit`.
>>> catalan_numbers(5)
[1, 1, 2, 5, 14, 42]
>>> catalan_numbers(2)
[1, 1, 2]
>>> catalan_numbers(-1)
Traceback (most recent call last):
ValueError: Limit for the Catalan sequence must be ≥ 0
"""
if upper_limit < 0:
raise ValueError("Limit for the Catalan sequence must be ≥ 0")
catalan_list = [0] * (upper_limit + 1)
# Base case: C(0) = C(1) = 1
catalan_list[0] = 1
if upper_limit > 0:
catalan_list[1] = 1
# Recurrence relation: C(i) = sum(C(j).C(i-j-1)), from j = 0 to i
for i in range(2, upper_limit + 1):
for j in range(i):
catalan_list[i] += catalan_list[j] * catalan_list[i - j - 1]
return catalan_list
if __name__ == "__main__":
print("\n********* Catalan Numbers Using Dynamic Programming ************\n")
print("\n*** Enter -1 at any time to quit ***")
print("\nEnter the upper limit (≥ 0) for the Catalan number sequence: ", end="")
try:
while True:
N = int(input().strip())
if N < 0:
print("\n********* Goodbye!! ************")
break
else:
print(f"The Catalan numbers from 0 through {N} are:")
print(catalan_numbers(N))
print("Try another upper limit for the sequence: ", end="")
except (NameError, ValueError):
print("\n********* Invalid input, goodbye! ************\n")
import doctest
doctest.testmod()
```