In this talk, we introduce a new formula for inversion of analytic functions. It is simple to use and it is easier to apply than the Lagrange-Burmann inversion formula, since no taking limits is required. By applying this formula we get some inversion formulas for certain functions. As a side result of this application, we also find new special numbers, which can also be defined using an analogue of the Pascal's triangle, and which have a simple relation with Bernoulli numbers. Using the formula for these numbers, we derive a new explicit formula for Bernoulli numbers.