%I #27 Jan 31 2024 10:54:59
%S 1,3,18,144,1368,14400,160992,1861632,21919104,260508672,3110985216,
%T 37241118720,446349219840,5352925446144,64215514275840,
%U 770468624990208,9244918222258176,110934787001942016,1331192054033547264,15974152308466384896,191688913661984243712
%N Non-crossing, non-nesting, 3-colored permutations on {1,2,...,n}.
%H Lily Yen, <a href="/A224992/b224992.txt">Table of n, a(n) for n = 0..99</a>
%H Wei Chen, <a href="http://summit.sfu.ca/item/14626">Enumeration of Set Partitions Refined by Crossing and Nesting Numbers</a>, MS Thesis, Department of Mathematics. Simon Fraser University, Fall 2014. Table 5.2, r=3.
%H Lily Yen, <a href="http://arxiv.org/abs/1211.3472">Crossings and Nestings for Arc-Coloured Permutations</a> and <a href="https://doi.org/10.46298/dmtcs.2339">Arc-coloured permutations</a>, PSAC 2013, Paris, France, June 24-28, Proc. DMTCS (2013) 743-754.
%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (20,-108,144).
%F G.f.: (1-17*x+66*x^2-36*x^3)/((1-2*x)*(1-6*x)*(1-12*x)).
%F a(n) = 9*2^n/20 +6^n/4 +12^n/20, n>0. - _R. J. Mathar_, Jun 11 2019
%e For n=3, a(3)= 144, the number of ways to color arcs of a permutation on {1,2,3} in 3 colors such that the arcs neither cross nor nest.
%t Join[{1}, LinearRecurrence[{20, -108, 144}, {3, 18, 144}, 20]] (* _Jean-François Alcover_, Jul 22 2018 *)
%o (PARI) Vec((1-17*x+66*x^2-36*x^3)/((1-2*x)*(1-6*x)*(1-12*x))+O(x^66)) \\ _Joerg Arndt_, Apr 24 2013
%K nonn,easy
%O 0,2
%A _Lily Yen_, Apr 24 2013