By Thomas Koshy

Just like the exciting Fibonacci and Lucas numbers, Catalan numbers also are ubiquitous. ''They have an analogous pleasant propensity for stoning up unexpectedly, really in combinatorial problems,'' Martin Gardner wrote in * medical American*. ''Indeed, the Catalan series is definitely one of the most often encountered series that remains imprecise sufficient to reason mathematicians missing entry to *Sloane's guide of Integer Sequences* to deplete inordinate quantities of strength re-discovering formulation that have been labored out lengthy ago,'' he persisted. As Gardner famous, many mathematicians may well be aware of the abc's of Catalan series, yet now not many are conversant in the myriad in their unforeseen occurrences, purposes, and houses; they crop up in chess forums, laptop programming, or even teach tracks. This e-book provides a transparent and finished advent to 1 of the really attention-grabbing subject matters in arithmetic. Catalan numbers are named after the Belgian mathematician Eugene Charles Catalan (1814-1894), who ''discovered'' them in 1838, even though he used to be now not the 1st individual to find them. the nice Swiss mathematician Leonhard Euler (1707-1763) ''discovered'' them round 1756, yet even earlier than then and notwithstanding his paintings was once now not identified to the surface global, chinese language mathematician Antu Ming (1692?-1763) first stumbled on Catalan numbers approximately 1730. Catalan numbers can be utilized by means of academics and professors to generate pleasure between scholars for exploration and highbrow interest and to sharpen numerous mathematical abilities and instruments, resembling trend popularity, conjecturing, proof-techniques, and problem-solving suggestions. This booklet isn't just meant for mathematicians yet for a miles higher viewers, together with highschool scholars, math and technology lecturers, machine scientists, and people amateurs with a modicum of mathematical interest. a useful source publication, it comprises an fascinating array of functions to desktop technology, summary algebra, combinatorics, geometry, graph thought, chess, and international sequence.

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N − 1)! = 2n + 1 2n −2 n+1 n+1 Since the RHS is an integer, it follows that every Catalan number Cn is also an integer; in other words, n + 1 | 2n n . 18 Catalan Numbers with Applications Wahlin’s Proof In 1911, G. E. Wahlin of the University of Illinois presented a proof that n+1 | 2n n . His proof, basically the same as Clarke’s, appeared as a solution to the problem proposed in 1910 by Feemster. 1 twice: 1 2n (2n)! n! = (2n)(2n − 1)(2n − 2) · · · (n + 1) (n + 1)! is an integer. 1, N = (2n)(2n−1)···(n+1) n!

A child prodigy, Erdös discovered negative numbers for himself at the age of three. At seventeen, he entered Eötvös University. Three years later, he wrote a delightful proof of Chebychev’s theorem that there is a prime number between the positive integers n and 2n. D. at the age of twenty-one. One of the most proliﬁc writers in mathematics, Erdös authored about 1500 articles and coauthored about 500. In a tribute in 1983, E. Straus described Erdös as “the prince of problem-solvers and the absolute monarch of problem-posers”Also called the “Euler of our time,”Erdös wrote extensively in number theory, combinatorics, the theory of functions, complex analysis, set theory, group theory, and probability.

M. Zerr of Temple College (now Temple University), Philadelphia. 3. 1 Prove that (nr)! )n is an integer for every n ≥ 0. (0)! (r)! r! = 1 are both integers, the result is true when n = 0 and n = 1. (kr)! ) k is an integer. Then (kr)! [(k + 1)r]! )k [(k + 1)r]! ÷ = · k+1 k k+1 (kr)! ) = (kr + r)(kr + r − 1) · · · (kr + 1) (k + 1)r! = (kr + r − 1) · · · (kr + 1) (r − 1)! = kr + r − 1 r−1 [(k+1)r]! ) k+1 is also an integer. Thus, by PMI, the result is true for every n ≥ 0. 16 Catalan Numbers with Applications It follows from this example that (nr)!