A225029 Non-crossing, non-nesting, 3-colored set partitions.

1, 4, 19, 103, 616, 3949, 26545, 184120, 1303135, 9341191, 67490044, 489978217, 3567727441, 26024391436, 190036459099, 1388593185079, 10150390743088, 74215146065461, 542704850311009, 3968914608295360, 29026988765886535, 212297824609934455, 1552734183515322436

Offset: 0

Author

Lily Yen, Apr 24 2013

Keywords

• nonn

Examples

			a(3) = 103 is the number of non-crossing, non-nesting, 3-colored set partitions on {1,2,3,4}.
		

Links

• Muniru A Asiru, Table of n, a(n) for n = 0..1145 (first 100 terms from Lily Yen)
• Eric Marberg, Crossings and nestings in colored set partitions, arXiv preprint arXiv:1203.5738 [math.CO], 2012-2013.
• Lily Yen, Crossings and Nestings for Arc-Coloured Permutations, arXiv:1211.3472 [math.CO], 2012-2013; and Arc-coloured permutations, PSAC 2013, Paris, France, June 24-28, Proc. DMTCS (2013) 743-754
• Lily Yen, Crossings and Nestings for Arc-Coloured Permutations and Automation, Electronic Journal of Combinatorics, 22(1) (2015), #P1.14.
• Index entries for linear recurrences with constant coefficients, signature(14,-59,74,-1).

Programs

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

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

Maple

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

Mathematica

LinearRecurrence[{14, -59, 74, -1}, {1, 4, 19, 103}, 23] (* Jean-François Alcover, Dec 14 2018 *)
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 *)

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

Formula

G.f.: (1 - 10*x + 22*x^2 - x^3)/(1 - 14*x + 59*x^2 - 74*x^3 + x^4).

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