  In \cite{ono}, K. Ono, S. Robins and P.T. Wahl considered the problem of
determining formulas for the number of representations of a natural number $n$
by a sum of $k$ triangular numbers and derived many applications, including the
one connecting these numbers with the number of representations of $n$ as a sum
of $k$ odd square integers. They also obtained an application to the number of
lattice points in the $k$-dimensional sphere. In this paper, we consider
triangular numbers with positive integer coefficients. First we show that if
the sum of these coefficients is a multiple of $8$, then the associated
generating function gives rise to a modular form of integral weight (when even
number of triangular numbers are taken). We then use the theory of modular
forms to get the representation number formulas corresponding to the triangular
numbers with coefficients. We also obtain several applications concerning the
triangular numbers with coefficients similar to the ones obtained in
\cite{ono}. In the second part of the paper, we consider more general mixed
forms (as done in Xia-Ma-Tian \cite{xia}) and derive modular properties for the
corresponding generating functions associated to these mixed forms. Using our
method we deduce all the 21 formulas proved in \cite[Theorem 1.1]{xia} and show
that our method of deriving the 21 formulas together with the $(p,k)$
parametrization of the generating functions of the three mixed forms imply the
$(p,k)$ parametrization of the Eisenstein series $E_4(\tau)$ and its
duplications. It is to be noted that the $(p,k)$ parametrization of $E_4$ and
its duplications were derived by a different method by K. S. Williams and his
co-authors. In the final section, we provide sample formulas for these
representation numbers in the case of 4 and 6 variable forms.
