  High temperature expansions for the susceptibility and the second correlation
moment of the classical N-vector model (also known as the O(N) symmetric
Heisenberg classical spin model or the as the lattice O(N) nonlinear sigma
model) on the square lattice are extended from order beta^{14} to beta^{21} for
arbitrary N. For the second field derivative of the susceptibility the series
expansion is extended from order beta^{14} to beta^{17}. For -2 < N < 2, a
numerical analysis of the series is performed in order to compare the critical
exponents gamma(N), nu(N) and Delta(N) to exact (though nonrigorous) formulas
and to compute the "dimensionless four point coupling constant" g_r(N). For N >
2, we present a study of the analiticity properties of chi, xi etc. in the
complex beta-plane and describe a method to estimate the parameters which
characterize their low-temperature behaviors. We compare our series estimates
to the predictions of the perturbative renormalization group theory, to exact
(but nonrigorous or conjectured) formulas and to the results of the 1/N
expansion, always finding a good agreement.
