The Relation of Tribinomial Coefficients with Triangular, Catalan and Mersenne Number - Wamiliana; Putri, Desiana; Notiragayu (Science and Technology Indonesia, 2026)
Numbers are inseparable from mathematics. Each of these types of numbers has its own distinct definition and properties. Numbers are not only used in mathematics, but are also essential in other fields such as philosophy, technology, and science. Tribinomial coefficients, Catalan numbers, and Mersenne numbers are three types of numbers that has their own uniqueness and beauty. Tribinomial coefficients derived from triangular number using similar definition for binomial coefficients. Triangular numbers constitute a class of figurative numbers derived from the systematic arrangement of discrete units (such as dots) into the geometric configuration of an equilateral triangle. The Catalan numbers constitute a sequence of positive integers that emerge in numerous combinatorial enumeration problems. Formally, the n-th Catalan number is defined by the closed-form expression: $C_n = \frac{1}{n+1} \binom{2n}{n}= \frac{(2n)!}{(n+1)!n!}, n \in N$. Mersenne numbers are the numbers known in mathematics that also have their own beauty and uniqueness. When represented using binary, all Mersenne numbers are repeating 1s. The first eight Mersenne numbers are 1, 3, 7, 15, 31, 63, 127, 255, which are represented in binary as 1, 11, 111, 1111, 11111, 111111, 1111111, 11111111. In this study the relationship of Tribinomial coefficients with Catalan and Marsenne numbers will be disccused.