Add Catalan number to maths (#6845)

* Add Catalan number to maths

* updating DIRECTORY.md

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maths/catalan_number.py

@@ -0,0 +1,51 @@
+"""
+
+Calculate the nth Catalan number
+
+Source:
+    https://en.wikipedia.org/wiki/Catalan_number
+
+"""
+
+
+def catalan(number: int) -> int:
+    """
+    :param number: nth catalan number to calculate
+    :return: the nth catalan number
+    Note: A catalan number is only defined for positive integers
+
+    >>> catalan(5)
+    14
+    >>> catalan(0)
+    Traceback (most recent call last):
+    ...
+    ValueError: Input value of [number=0] must be > 0
+    >>> catalan(-1)
+    Traceback (most recent call last):
+    ...
+    ValueError: Input value of [number=-1] must be > 0
+    >>> catalan(5.0)
+    Traceback (most recent call last):
+    ...
+    TypeError: Input value of [number=5.0] must be an integer
+    """
+
+    if not isinstance(number, int):
+        raise TypeError(f"Input value of [number={number}] must be an integer")
+
+    if number < 1:
+        raise ValueError(f"Input value of [number={number}] must be > 0")
+
+    current_number = 1
+
+    for i in range(1, number):
+        current_number *= 4 * i - 2
+        current_number //= i + 1
+
+    return current_number
+
+
+if __name__ == "__main__":
+    import doctest
+
+    doctest.testmod()