constructing the function – how to perform a conv2 equivalent conv2 of two vectors in Mathematica?

I have a matlab code

filter = 1;
F = conv2 (double ([1 2 1]),double([1 2 1]& # 39;)) / 16;
for i = 1: some_integer
filter = conv2 (double (filter), double (F));
end

in the code F = conv2 (double ([1 2 1]),double([1 2 1]& # 39;)) / 16; equals
a 3 x 3 matrix {{0,0625, 0,125, 0,0625}, {0,125, 0,25, 0,125}, {0,0625, 0,125, 0,0625}}

the filter changes the value of the first iteration from 1 to F and then, successive iterations, the result (convolution matrix) becomes larger and larger.

I have trouble making this convolution in Mathematica, despite the review of the documentation. An idea of ​​how one could make such a convolution?

my attempt below to calculate F before the loop seems incorrect.

F = Flatten[(ListConvolve[{{0, 0, 0}, {1, 2, 1}, {0, 0, 0}}, {{1, 2, 1}, {1, 2, 1}, {1, 2, 1}}, {-1, 1}, g, Plus, List]/16.0), 2]