I needed to generate stirling numbers programmatically but couldn't find a proper implementation available in python. Being a non-mathematician, it was quite hard to understand and thus I wrote this script. From my testing, the outputs are correct.

stirling_number.py

import functools

_use_cache = True


def set_use_cache(use_cache):
    global _use_cache
    _use_cache = bool(use_cache)


def optional_cache(func):
    @functools.wraps(func)
    def wrapper(*args, **kwargs):
        if not _use_cache:
            return func(*args, **kwargs)
        key = ''
        if args:
            key += '@'.join(map(str, args))
        if kwargs:
            kwarg_names = list(kwargs.keys())
            kwarg_names.sort()
            if key:
                key += '#'
            key += '@'.join(['{}={}'.format(kwarg_key, kwargs[kwarg_key]) for kwarg_key in kwarg_keys])

        if not getattr(func, 'cache', None):
            setattr(func, 'cache', dict())
        if key not in func.cache:
            func.cache[key] = func(*args, **kwargs)
        return func.cache[key]
    return wrapper

@optional_cache
def first_kind(n, k):
    if k == 0:
        return 0
    elif n == k:
        return int(n != 0)
    return first_kind(n - 1, k - 1) + (n - 1) * first_kind(n - 1, k)


@optional_cache
def signed_first_kind(n, k):
    return first_kind(n, k) * pow(-1, n - k)


@optional_cache
def second_kind(n, k):
    if n == 0 and k == 0:
        return 1 
    elif n == 0 or k == 0:
        return 0
    return second_kind(n - 1, k - 1) + k * second_kind(n - 1, k)


if __name__ == '__main__':
    import sys

    if len(sys.argv) < 3:
        print("Usage:")
        print("{} {} <n:int> <k:int>".format(sys.executable, sys.argv[0]))
        sys.exit(1)

    def _get_int(value, name):
        try:
            return int(value.strip())
        except ValueError:
            print("argument {} is not an int: {}".format(n, value))
            sys.exit(1)
    
    n = _get_int(sys.argv[1].strip(), "n")
    k = _get_int(sys.argv[2].strip(), "k")
    
    if n <= 0:
        print("n > 0 required")
        sys.exit()

    print("First kind unsigned: " + str(first_kind(n, k)))
    print("First kind signed: " + str(signed_first_kind(n, k)))
    print("Second kind: " + str(second_kind(n, k)))