This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A183758 #11 Jul 25 2021 02:38:17 %S A183758 1,1,2,5,12,44,176,809,3883,19265 %N A183758 Unlabeled super-Catalan numbers: patterns of nonintersecting chords joining unlabeled points on a flippable circle. %C A183758 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. %e A183758 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. %Y A183758 Cf. A001003 super-Catalan numbers. A183757 similar but counts reflections. A183760 this sequence decomposed by number of chords in the circles. %K A183758 nonn,more %O A183758 0,3 %A A183758 _David Scambler_, Jan 06 2011