Properties of coefficients in some superpositions of generating functions

The generating function $ \ln((1-F(x))^{-1})$ where $F(x)$ is an ordinary generating function with the integer coefficients is considered. Some properties ot its coefficients allowing the construction of probabilistic primality tests are obtained. The connection of them with the existing primality tests is shown. Some new properties of Lucas numbers and binomial coefficients $2n-1\choose n-1$ are obtained too.