  Following Lucas and then other Fibonacci people Kwasniewski had introduced
and had started ten years ago the open investigation of the overall F-nomial
coefficients which encompass among others Binomial, Gaussian and Fibonomial
coefficients with a new unified combinatorial interpretation expressed in terms
of cobweb posets' partitions and tilings of discrete hyperboxes. In this paper,
we deal with special subfamily of T-nomial coefficients.
  The main aim of this note is to develop the theory of T-nomial coefficients
with the help of generating functions. The binomial-like theorem for T-nomials
is delivered here and some consequences of it are drawn. A new combinatorial
interpretation of T-nomial coefficients is provided and compared with the
Konvalina way of objects' selections from weighted boxes. A brief summary of
already known properties of T-nomial coefficients is served.
