How can I generate a Toeplitz matrix in the correct form for performing discrete convolution?

Discrete convolution can be performed via the Toeplitz matrix, as shown below (Wiki article):

Note that this is not the exact same form as as the general Toeplitz matrix, but it has experienced various shifts and zero-paddings.

Is there a way to achieve this in numpy purely based on roll, hstack etc., i.e. without using any for loops? I have tried all sorts of shifts but I can't really get it in to the form shown above.

Yes, you can use scipy.linalg.toeplitz:

import numpy as np
from scipy import linalg

h = np.arange(1, 6)

padding = np.zeros(h.shape[0] - 1, h.dtype)
first_col = np.r_[h, padding]
first_row = np.r_[h[0], padding]

H = linalg.toeplitz(first_col, first_row)

print(repr(H))
# array([[1, 0, 0, 0, 0],
#        [2, 1, 0, 0, 0],
#        [3, 2, 1, 0, 0],
#        [4, 3, 2, 1, 0],
#        [5, 4, 3, 2, 1],
#        [0, 5, 4, 3, 2],
#        [0, 0, 5, 4, 3],
#        [0, 0, 0, 5, 4],
#        [0, 0, 0, 0, 5]])