We consider the generating series of Euler numbers of Hilbert schemes of points on a singular plane curve. After a change of variables, we show this counts the number of h-nodal curves in deformations in nearby h-parameter families. Similar arguments, made globally, lead to a proof of Gottsche's conjecture on the universality of numbers of nodal curves on surfaces. Finally, we conjecturally relate a refinement of this generating series to the homological knot invariants of the links of the singularities.