Sample of generating functions implementation using Sympy.

gen_func_sample.py

"""
Problem:
    Cody has 4 types of onions.
    The number of purple onions can be any non-negative integer.
    The number of green onions is a multiple of 2.
    The number of red onions is a multiple of 3.
    The number of blue onions is a multiple of 5.
    If Cody has N onions, how many different distributions of colors can there be?
"""

from sympy import poly
from sympy.abc import x

R = poly(1, x)
P = [poly(0, x)] * 4
M = [1, 2, 3, 5]
N = 1000

for i in range(4):
    for k in range(0, N//M[i]+1):
        P[i] += poly(x**(M[i]*k), x)
    R *= P[i]

print("Answer:", R.coeffs()[N])