We present two novel approaches for the computation of the exact distribution of a pattern in a long sequence. Both approaches take into account the sparse structure of the problem. The first approach relies on a partial recursion computing the largest eigenvalue of the the transition matrix of a Markov chain embedding. The second approach uses fast Taylor expansions of an exact bivariate rational reconstruction of the distribution. We illustrate the interest of both approaches on a simple toy-example and two biological applications: the transcription factors of the Human Chromosome 5 and the PROSITE signatures of functional motifs in proteins. On these examples our methods demonstrate their complementarity and their hability to extend the domain of feasibility for exact computations in pattern problems to a new level.