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 A363964 #12 Sep 06 2023 21:03:13 %S A363964 3,14,55,195,644,2016,6048,17520,49280,135168,362752,955136,2472960, %T A363964 6307840,15876096,39481344,97124352,236584960,571146240,1367539712, %U A363964 3249799168,7669284864,17983078400,41916825600,97165246464,224076496896,514272002048,1174992322560 %N A363964 Number of unordered pairs of non-intersecting non-self-intersecting paths, singletons included, with nodes that cover all vertices of a convex labeled n-gon. %C A363964 For each such path there is a sequence of distinct vertices of the n-gon, each (except the last one) connected by a segment with the next vertex in the sequence; the segments have no common internal points. The path itself is the union of the set of these segments and is thus direction-independent: reversing the order of the vertices leads to the same path. If the sequence of vertices has length 1 then there are no segments; we call such a path a singleton. %F A363964 a(n) = n*(n-1)*(n^2+n+36)*2^(n-8)/3. %e A363964 a(4)=14 since if one of the paths is a singleton (4 choices), then there are A001792(3)=3 choices for the other path, and otherwise for the two paths there are A308914(4)=2 choices, so a(4)=4*3+2=14. %Y A363964 Cf. A001792, A308914, A332426, A360716, A360717. %K A363964 nonn %O A363964 3,1 %A A363964 _Ivaylo Kortezov_, Jun 30 2023