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A183758 Unlabeled super-Catalan numbers: patterns of nonintersecting chords joining unlabeled points on a flippable circle 2
1, 1, 2, 5, 12, 44, 176, 809, 3883, 19265 (list; graph; refs; listen; history; text; internal format)



Interpret the super-Catalan sequence (A001003) as follows:  Ways to insert parentheses in a string of n+1 symbols. The parentheses must be balanced but there is no restriction on the number of pairs of parentheses. The number of letters inside a pair of parentheses must be at least 2. Parentheses enclosing the whole string are ignored.  Now picture the x's and parentheses as equally spaced unlabeled points on a circle with chords joining paired parentheses and x's having no chord. Circles thus produced may have n+1, n+3, ..., 3*n-1 points and up to n-1 chords. The circle may be rotated and flipped over. a(n) is the count of unique patterns.


Table of n, a(n) for n=0..9.


A183757(3) = 6: (xx)xx, (xx)(xx), (xxx)x, ((xx)x)x, x(x(xx)), xxxx and removing the reflected pattern x(x(xx)) leaves a(3) = 5.


Cf. A001003 super-Catalan numbers. A183757 similar but counts reflections. A183760 this series decomposed by number of chords in the circles.

Sequence in context: A215789 A127137 A172239 * A334811 A071787 A343813

Adjacent sequences:  A183755 A183756 A183757 * A183759 A183760 A183761




David Scambler, Jan 06 2011



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Last modified June 18 20:45 EDT 2021. Contains 345121 sequences. (Running on oeis4.)