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. a(n) is the count of unique patterns.
EXAMPLE
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.
CROSSREFS
KEYWORD
nonn,more
AUTHOR
David Scambler, Jan 06 2011
STATUS
approved