A183758 Unlabeled super-Catalan numbers: patterns of nonintersecting chords joining unlabeled points on a flippable circle.

1, 1, 2, 5, 12, 44, 176, 809, 3883, 19265

Offset

0,3

Comments

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.

Links

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

Example

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.

Crossrefs

Cf. A001003 super-Catalan numbers. A183757 similar but counts reflections. A183760 this sequence 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

Keyword

nonn,more

Author

David Scambler, Jan 06 2011

Status

approved