%I
%S 1,1,2,6,17,74,324,1558,7640,38245
%N Unlabeled superCatalan numbers: patterns of nonintersecting chords joining unlabeled points on a circle
%C Interpret the superCatalan 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*n1 points and up to n1 chords. The circle may be rotated. a(n) is the count of unique patterns.
%e superCatalan(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.
%Y Cf. A001003 superCatalan numbers. A183758 similar but with reflections discounted. A183759 this series decomposed by number of chords in the circles.
%K nonn
%O 0,3
%A _David Scambler_, Jan 06 2011
