A183758 -id:A183758 - cp's OEIS frontend

A183757 Unlabeled super-Catalan numbers: patterns of nonintersecting chords joining unlabeled points on a circle.

1, 1, 2, 6, 17, 74, 324, 1558, 7640, 38245

Offset: 0

David Scambler, Jan 06 2011

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. a(n) is the count of unique patterns.

			super-Catalan(3) = 11: (xx)xx, x(xx)x, xx(xx), (xx)(xx), (xxx)x, x(xxx),((xx)x)x, (x(xx))x, x((xx)x), x(x(xx)), xxxx. This sequence counts unique patterns up to rotation so  a(3) = 6: (xx)xx, (xx)(xx), (xxx)x, ((xx)x)x, x(x(xx)), xxxx.

Cf. A001003 super-Catalan numbers. A183758 similar but with reflections discounted. A183759 this sequence decomposed by number of chords in the circles.

A183760 Unlabeled super-Catalan numbers: patterns of nonintersecting chords joining unlabeled points on a flippable circle, triangle decomposed by number of chords.

Original entry on oeis.org

1, 1, 1, 1, 2, 2, 1, 2, 5, 4, 1, 3, 10, 18, 12, 1, 3, 16, 51, 71, 34, 1, 4, 26, 116, 271, 280, 111, 1, 4, 36, 233, 776, 1349, 1124, 360, 1, 5, 53, 421, 1908, 4749, 6487, 4415, 1226

Offset: 1

David Scambler, Jan 06 2011

Decomposition of A183758 by number of chords (pairs of parentheses). Row sums are A183758. See A183758 for description.
Triangle is
1
1 1
1 2  2
1 2  5  4
1 3 10 18 12
1 3 16 51 71 34
...
Right diagonal is apparently A001895.

Decomposition of A183758. Cf. analogous decomposition of super-Catalan numbers in triangle A033282.

cp's OEIS Frontend

A183757 Unlabeled super-Catalan numbers: patterns of nonintersecting chords joining unlabeled points on a circle.

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