%I #47 Jan 31 2024 10:32:37
%S 1,4,19,103,616,3949,26545,184120,1303135,9341191,67490044,489978217,
%T 3567727441,26024391436,190036459099,1388593185079,10150390743088,
%U 74215146065461,542704850311009,3968914608295360,29026988765886535,212297824609934455,1552734183515322436
%N Non-crossing, non-nesting, 3-colored set partitions.
%H Muniru A Asiru, <a href="/A225029/b225029.txt">Table of n, a(n) for n = 0..1145</a> (first 100 terms from Lily Yen)
%H Eric Marberg, <a href="http://arxiv.org/abs/1203.5738">Crossings and nestings in colored set partitions</a>, arXiv preprint arXiv:1203.5738 [math.CO], 2012-2013.
%H Lily Yen, <a href="http://arxiv.org/abs/1211.3472">Crossings and Nestings for Arc-Coloured Permutations</a>, arXiv:1211.3472 [math.CO], 2012-2013; 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 Lily Yen, <a href="https://doi.org/10.37236/4080">Crossings and Nestings for Arc-Coloured Permutations and Automation</a>, Electronic Journal of Combinatorics, 22(1) (2015), #P1.14.
%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature(14,-59,74,-1).
%F G.f.: (1 - 10*x + 22*x^2 - x^3)/(1 - 14*x + 59*x^2 - 74*x^3 + x^4).
%F a(n) = 14*a(n-1) -59*a(n-2) +74*a(n-3) -a(n-4), with a(0) = 1, a(1) = 4, a(2) = 19 and a(3) = 103. - _Muniru A Asiru_, Dec 18 2018
%e a(3) = 103 is the number of non-crossing, non-nesting, 3-colored set partitions on {1,2,3,4}.
%p seq(coeff(series((1-10*x+22*x^2-x^3)/(1-14*x+59*x^2-74*x^3+x^4),x,n+1), x, n), n = 0 .. 25); # _Muniru A Asiru_, Dec 18 2018
%t LinearRecurrence[{14, -59, 74, -1}, {1, 4, 19, 103}, 23] (* _Jean-François Alcover_, Dec 14 2018 *)
%t CoefficientList[Series[(1 - 10 x + 22 x^2 - x^3) / (1 - 14 x + 59 x^2 - 74 x^3 + x^4), {x, 0, 25}], x] (* _Vincenzo Librandi_, Dec 20 2018 *)
%o (PARI) Vec((1-10*x+22*x^2-x^3)/(1-14*x+59*x^2-74*x^3+x^4)+O(x^66)) \\ _Joerg Arndt_, Apr 24 2013
%o (GAP) a:=[1,4,19,103];; for n in [5..25] do a[n]:=14*a[n-1]-59*a[n-2]+74*a[n-3]-a[n-4]; od; a; # _Muniru A Asiru_, Dec 18 2018
%o (Magma) I:=[1,4,19,103]; [n le 4 select I[n] else 14*Self(n-1)-59*Self(n-2)+74*Self(n-3)-Self(n-4): n in [1..30]]; // _Vincenzo Librandi_, Dec 20 2018
%K nonn
%O 0,2
%A _Lily Yen_, Apr 24 2013