  We show that the ultradifferentiable-like classes of smooth functions
introduced and studied by S. Pilipovi\'c, N. Teofanov and F. Tomi\'c are
special cases of the general framework of spaces of ultradifferentiable
functions defined in terms of weight matrices in the sense of A. Rainer and the
third author. We study classes "beyond geometric growth factors" defined in
terms of a weight sequence and an exponent sequence, prove that these new types
admit a weight matrix representation and transfer known results from the
matrix-type to such a non-standard ultradifferentiable setting.
