Large N limit of O(N) vector models

Using a simple identity between various partial derivatives of the energy of the vector model in 0+0 dimensions, we derive explicit results for the coefficients of the large N expansion of the model. These coefficients are functions in a variable $\rho^2$, which is the expectation value of the two point function in the limit $N=\infty$. These functions are analytic and have only one (multiple) pole in $\rho^2$. We show to all orders that these expressions obey a given general formula. Using this formula it is possible to derive the double scaling limit in an alternative way. All the results obtained for the double scaling limit agree with earlier calculations. (to be published in Physics Letters B)