Math Project: Catalan Sequence

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Catalan Numbers, 1,1,2,5,14,42,...
Introduction
The standard Catalan sequence 1,1,2,5,14,42,.... was developed by Eugene Charles Catalan, Belgian mathematician, and comprises a sequence of positive integers. These integers are unique and occur in diverse combinatorial applications and situations, including triangulations, trees, and lattice paths, among other mathematical problems (Feng and Bai-Ni 56). The essay sets out to research Catalan Numbers, 1,1,2,5,14,42,....The paper will review the history, combinatorial formulas, and various applications and interpretations of Catalan numbers.
History of Catalan Numbers
In 1730, Ming Antu, a Mongolian mathematician ,wrote a mathematical book in which he used a Catalan sequence to express sin(2α) and sin(4α) in terms of sin(α) (Feldman 1657). He also obtained a recurrence formula.
Later, the Catalan numbers were redefined by Leonhard Euler ,who introduced the term, “triangulation.” He referred to the sequence as polygon triangulations with n+2 vertices (Gunnells). In his mathematical definition, he devised the first nine terms as used in the Catalan sequence today. Through random guessing, he developed the first Catalan Formula as follows:
In 1758, a Hungarian mathematician developed the Catalan sequence into a combinatory recurrence. He provided evidence and proof that Catalan numbers were combinatory using the following formula
Eugene Charles Catalan defined the modern-day standard Catalan formulas in 1838. He devised the standard Catalan formula as follows:
The Catalan Sequence Formulas
1 Combinatorial Interpretations
The combinatorial interpretation makes use of mathematical brackets. They are defined as
Brackets
Sequence example where n = 3
Catalan numbers are used to solve combinatorial problems such as correctly matching n pairs of brackets (Gunnells). Assuming a string comprising of n pairs of brackets, the number of techniques by which 2 or more brackets can be added to get an n+2 string that exactly matches the brackets is derived using the Catalan numbers. The two strings can be thought of as comprising two valid substrings, where one is enclosed in a new pair of brackets (Feldman 1645).
When all the possible values (outcomes) of i are considered, Segner’s recurrence formula is
derived as follows.
Diverse combinatorial interpretations can be summed up into brackets problems when one-to-one correspondences are defined. Here are the different combinatorial interpretations of Catalan numbers:
Interpretations of Catalan numbers
Applications for Catalan numbers
1 Triangulations
Polygon triangulation consists of a set of non-intersecting diagonals. Polygon triangulation occurrences can be defined using n+2 vertices and are represented by the nth Catalan number (Gunnells).
Polygon triangulations
2 Lattice paths
The nth Catalan number represents the total number ...