Question 1099746: What is the next number in the series 1,1,2,5,14,42?
Answer by richwmiller(17219)   (Show Source):
You can put this solution on YOUR website! The Catalan numbers (1, 2, 5, 14, 42, 132, 429, 1430, 4862, 16796, 58786, 208012, 742900, 2674440, 9694845, ...), named after Eugène Charles Catalan (1814–1894), arise in a number of problems in combinatorics.
Among other things, the Catalan numbers describe:
the number of ways a polygon with n+2 sides can be cut into n triangles
the number of ways to use n rectangles to tile a stairstep shape (1, 2, ..., n−1, n).
the number of ways in which parentheses can be placed in a sequence of numbers to be multiplied, two at a time
the number of planar binary trees with n+1 leaves
the number of paths of length 2n through an n-by-n grid that do not rise above the main diagonal
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