%I #13 Mar 12 2023 14:18:14
%S 1,1,3,12,55,273,1372,6824,33489,162405,779801,3713436,17560803,
%T 82553597,386105790,1797803248,8338313697,38539754649,177581276639,
%U 815982230060,3740047627071,17103604731961,78054858200448,355541644914072,1616688603539025
%N Number of noncrossing caterpillars with n edges.
%C A noncrossing caterpillar is a noncrossing tree that is a caterpillar tree (also called a caterpillar graph).
%C The number of nodes is n + 1. All trees up to 5 edges (or 6 nodes) are caterpillars.
%H Andrew Howroyd, <a href="/A361356/b361356.txt">Table of n, a(n) for n = 0..1000</a>
%H Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/CaterpillarGraph.html">Caterpillar Graph</a>.
%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Caterpillar_tree">Caterpillar tree</a>.
%H <a href="/index/Rec#order_08">Index entries for linear recurrences with constant coefficients</a>, signature (12,-52,104,-114,76,-32,8,-1).
%F a(n) = 12*a(n-1) - 52*a(n-2) + 104*a(n-3) - 114*a(n-4) + 76*a(n-5) - 32*a(n-6) + 8*a(n-7) - a(n-8) for n >= 8.
%F G.f.: (1 - 11*x + 43*x^2 - 76*x^3 + 77*x^4 - 37*x^5 + 6*x^6)/((1 - x)^2*(1 - 5*x + 3*x^2 - x^3)^2).
%F G.f.: 1 + x + x^2*(3 - 2*x + (4 - 3*x + x^2)*F(x) + (1 + 2*x)*F(x)^2)/(1 - x)^2 where F(x) = x*(2 - x)/(1 - 5*x + 3*x^2 - x^3).
%o (PARI) Vec((1 - 11*x + 43*x^2 - 76*x^3 + 77*x^4 - 37*x^5 + 6*x^6)/((1 - x)^2*(1 - 5*x + 3*x^2 - x^3)^2) + O(x^30))
%Y Row sums of A361357.
%Y Cf. A001764 (noncrossing trees), A005418 (unlabeled), A245012 (labeled).
%Y Cf. A361358, A361359 (up to rotations), A361360 (up to rotations and reflections).
%K nonn,easy
%O 0,3
%A _Andrew Howroyd_, Mar 09 2023