  We study the counting function of topological Poincar\'e series associated
with rational homology sphere plumbed 3-manifold with connected negative
definite tree, interpreting as an alternating sum of coefficient functions
associated with some Taylor expansions. It is motivated by a theorem of Szenes
and Vergne which expresses these coefficient functions in terms of
Jeffrey--Kirwan residues. This is used to prove the uniqueness of the
quasipolynomiality inside a special cone, the structure of the counting
function in terms of the graph and construction for a polynomial generalization
of the Seiberg--Witten invariant given by the Poincar\'e series. We also
reprove and discuss surgery formulas of N\'emethi for the counting function,
and of Braun and N\'emethi for the Seiberg--Witten invariant.
