  According to the G\"ottsche conjecture (now a theorem), the degree N^{d,
delta} of the Severi variety of plane curves of degree d with delta nodes is
given by a polynomial in d, provided d is large enough. These "node
polynomials" N_delta(d) were determined by Vainsencher and Kleiman-Piene for
delta <= 6 and delta <= 8, respectively. Building on ideas of Fomin and
Mikhalkin, we develop an explicit algorithm for computing all node polynomials,
and use it to compute N_delta(d) for delta <= 14. Furthermore, we improve the
threshold of polynomiality and verify G\"ottsche's conjecture on the optimal
threshold up to delta <= 14. We also determine the first 9 coefficients of
N_delta(d), for general delta, settling and extending a 1994 conjecture of Di
Francesco and Itzykson.
