  In this paper we study the derivation of a certain type of NLS from many-body
interactions of bosonic particles. We consider a model with a finite linear
combination of $n$-body interactions, where $n \geq 2$ is an integer. We show
that the $k$-particle marginal density of the BBGKY hierarchy converges when
particle number goes to infinity, and the limit solves a corresponding infinite
Gross-Pitaevskii hierarchy. We prove the uniqueness of factorized solution to
the Gross-Pitaevskii hierarchy based on a priori space time estimates. The
convergence is established by adapting the arguments originated or developed in
\cite{ESY}, \cite{KSS} and \cite{CPquintic}. For the uniqueness part, we expand
the procedure followed in \cite{KM} by introducing a different board game
argument to handle the new contraction operator. This new board game argument
helps us obtain a good estimate on the Duhamel terms. In \cite{KM}, the
relevant space time estimates are assumed to be true, while we give a prove for
it.
