  In two papers, Little and Sellers introduced an exciting new combinatorial
method for proving partition identities which is not directly bijective.
Instead, they consider various sets of weighted tilings of a $1 \times \infty$
board with squares and dominoes, and for each type of tiling they construct a
generating function in two different ways, which generates a $q$-series
identity. Using this method, they recover quite a few classical $q$-series
identities, but Euler's Pentagonal Number Theorem is not among them. In this
paper, we introduce a key parameter when constructing the generating functions
of various sets of tilings which allows us to recover Euler's Pentagonal Number
Theorem along with an infinite family of generalizations.
