How can I manually convolve the laplacian kernel?

I've been reading that the Laplacian operator is the sum of two "second order derivatives", Ixx and Iyy.

But I've also seen several places where the Laplacian kernel is listed as one of following:

Because convolution is associative, I've been testing out creating my own Laplacian kernel via the following:

laplaceKern = conv2(xkern, xkern) + conv2(ykern, ykern)

But this produces a kernel as so:

I can't seem to arrive at either of the aforementioned accepted laplacian kernels via convolution. What am I missing?