The paper introduces the concept of generalized Catalan numbers as a useful tool for solving many theoretical and applied combinatorial-probabilistic problems. Combined with analytical transformation software algorithms, the generalized Catalan numbers simplify the solution of many problems in computer science and applied mathematics. In particular, they turn out to be an effective tool for solving problems related to the registration of random-dot images, in signal transformations of various degrees of smoothness, and when developing speed-optimal algorithms for searching for pulsed-point objects with a random generation time of super short pulses. The proposal of the authors to formulate the problems of enumerative combinatorics in a word-symbolic form naturally leads to multidimensional extensions of the classical Catalan numbers and has several advantages. The combined use of multidimensional Catalan numbers and high-performance computer algebra systems enables solving a number of complex applied problems related to the reliability of the registration of random-dot images.