Severi Degrees in Cogenus 4

In this paper, we give closed-form formulae for Severi degrees in cogenus 3 and 4 using Ran's method. These formulae coincide with those of I. Vainsencher and for cogenus 3 case, that of J. Harris and R. Pandharipande. Another result of this paper is that we calculate the degree of the polynomial $N(\pi,\delta,d)$ in $d$, which is the degree of the locus of curves $C$ in P^2 with degree d having $\delta$ nodes and $C \cap L$ is of type $\pi$ to fixed line L. Using this result, we also calculate the coefficients of two leading terms of Severi polynomial, $N(\delta,d)$.