Consider the set of 4 x 4 matrices with integer entries of a fixed determinant n. The group GL(4, \Z) acts on the right by multiplication. Similarly, the symmetric group S_4 acts on the left via multiplication by permutation matrices. The entry a_n is the number of elements in the double orbit space S_4\det^{-1}(n)/GL(4,\Z). The sequence a_n also counts the number of isomorphism classes of simplicial cones in \Z^4 of a certain index, or alternatively the number of affine toric varieties in dimension 4 arising from simplicial cones.
For n = 2, three representatives are [4,4]((1,0,0,0),(0,1,0,0),(0,0,1,0),(0,0,1,2)), ((1,0,0,0),(0,1,0,0),(0,0,1,0),(0,1,1,2)) and ((1,0,0,0),(0,1,0,0),(0,0,1,0),(1,1,1,2)).
