![]() Cambridge,Įngland: Cambridge University Press, 2003. Computationalĭiscrete Mathematics: Combinatorics and Graph Theory with Mathematica. "A Determinant Property of Catalan Numbers." Disc. "Correspondences Between Plane Trees and Binary Sequences." "Catalan Numbers, Their Generalization, and Their Uses." Math. "Dissecting a Polygon Into Triangles."īull. Mathematics: A Foundation for Computer Science, 2nd ed. Research Bibliography of Two Special Number Sequences, 6th ed. Sequence that Materializes in Unexpected Places." Sci. ![]() Travel and Other Mathematical Bewilderments. "Catalan Strikes Again! How Likely is a Function to be Convex?" Math. Great Problems of Elementary Mathematics: Their History and Solutions. "Euler's Problem of Polygon Division." §7 in 100 "A New Combinatorial Interpretation for GeneralizedĬatalan Numbers." Disc. "The Computation of Catalan Numbers." Math. and Bailey,īy Experiment: Plausible Reasoning in the 21st Century. "Prime and Prime Power Divisibility of Catalan Numbers." "Some Remarks and Results on Catalan Numbers." Proc. ![]() Internal nodes, the number of rooted plane bushes with graph edges, the number of extendedīinary trees with internal nodes, and the number of mountains which can be drawnĭownstrokes, the number of noncrossing handshakes possible across a round table betweenĪ generalization of the Catalan numbers is defined by Strokes, the number of ways of forming an -fold exponential, the number of rooted planar binary trees The number of states possible in an - flexagon, the number of differentĭiagonals possible in a frieze pattern with rows, the number of Dyck The ballot problem, the number of trivalent planted planar trees (Dickau illustrated above), Letters ( Catalan's problem), the solution to The Catalan numberĪlso gives the number of binary bracketings The Catalan numbers turn up in many other related types of problems. So 5 is the last digit for all up to at least with the exception of 1, 3, 5, 7, and 8. In fact, the last digits of the odd Catalan numbers are 1, 5, 9, 5, 9,ĥ, 9, 7, 5, 5, 5, 5, 5. Rare for a long sequence of essentially random base-5 digits to contain only in 0,ġ, and 2. (OEIS A038003).Įnd in 5 unless the base-5 expansion of uses only the digits 0, 1, 2, so it would be extremely The only odd Catalan numbers are those of the form. (Of course, much more than thisĬan be said about the factorization of. Therefore, is the largest Catalan prime, making and the only Catalan primes. The above recurrence relation gives the Catalanįrom the definition of the Catalan number, every prime divisor of is less than.
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