How to do inverse Z transform with python, this function will break up the partial fraction assuming Z transform. Note that it doesn't handle complex roots, if you have complex roots use the scipy residuez function to find the poles and coefficients.

Inverse_Z_transform_python.py

### Solving Z domain partial fraction with sympy 
from sympy import *
z = symbols('z')

def inverse_z(h):
    '''finds X(z)/z for the inverse z transform table look up
    '''
    h = h/z
    h = h.apart() * z
    return expand(h)


n = z**2 + z + (z/(z-1))
d = z**2 - 3*z/2 + 1/2
a = n/d
a = inverse_z(a)
print(a)
#a
# a.cancel().apart()
'''
Note, whenever you try to do partial fractions for z inv, you must first divide the system by z
    and then multiply the partial fraction result by z. So you perform partials on X(z)/z = LHS/z, and answer after partial will be multiplied by z so that your final answer is X(z)
'''