A216947 - OEIS

%I #74 Oct 31 2024 06:49:40

%S 1,1,3,11,47,225,1173,6529,38265,233795,1478265,9619065,64135965,

%T 436693431,3028075677,21335364345,152466207953,1103356966359,

%U 8075335006933,59706844371261,445545262970505,3352776605782479,25424338834654587

%N Number of 2-colored non-crossing set partitions of [n].

%H Andrew Howroyd, <a href="/A216947/b216947.txt">Table of n, a(n) for n = 0..200</a>

%H Alin Bostan, Jordan Tirrell, Bruce W. Westbury and Yi Zhang, <a href="https://arxiv.org/abs/1911.10288">On sequences associated to the invariant theory of rank two simple Lie algebras</a>, arXiv:1911.10288 [math.CO], 2019.

%H Alin Bostan, Jordan Tirrell, Bruce W. Westbury and Yi Zhang, <a href="https://arxiv.org/abs/2110.13753">On some combinatorial sequences associated to invariant theory</a>, arXiv:2110.13753 [math.CO], 2021.

%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. See Th. 3.4.1.

%H Juan B. Gil, David Kenepp and Michael Weiner, <a href="https://permutationpatterns.com/files/2020/06/WednesdayA_Gil.pdf">Pattern-avoiding permutations by inactive sites</a>, Pennsylvania State University, Altoona (2020).

%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.

%F Marberg gives a recurrence and g.f.

%F a(n) = (2*(5*n^2 + 6*n - 2)*a(n-1) - 9*(n - 2)*(n + 1)*a(n-2))/((n + 2)*(n + 3)) for n >= 2. - _Andrew Howroyd_, Dec 26 2019, after Marberg theorem 1.7.

%F a(n) ~ 3^(2*n + 11/2) / (16*Pi*n^4). - _Vaclav Kotesovec_, Jan 03 2020

%F a(n) = Sum_{j=0..n} Sum_{k=0..j} C(n-1,j)*C(j+2,k) * C(j+2,k+1) * C(j+2,k+2) / (C(j+2,1) * C(j+2,2)). - _Michael D. Weiner_, Jan 21 2020

%F G.f.: hypergeom([1/4, 5/4],[2],-64*x/((9*x-1)^3*(x-1)))/(3*(1-9*x)^(3/4)*x^2*(1-x)^(1/4))-(2*x-1)*(x-1)/(3*x^2). - _Mark van Hoeij_, Jul 27 2021

%F a(n) = ((n+3)*hypergeom([1/2, -1-n, -2-n],[2, 2],4)-3*(n+2)*hypergeom([1/2, -1-n, -n],[2, 2],4))/(2*n*(n+2)) for n > 0. - _Mark van Hoeij_, Nov 06 2023

%p A216947 := proc(n)

%p     option remember ;

%p     local npr  ;

%p     if n <=1 then

%p         1 ;

%p     else

%p         npr := n-2 ;

%p         9*npr*(npr+3)*procname(n-2)-2*(5*npr^2+26*npr+30)*procname(n-1) ;

%p         -%/(npr+4)/(npr+5) ;

%p     end if;

%p end proc:

%p seq(A216947(n),n=0..20) ; # _R. J. Mathar_, Nov 16 2012

%t a[n_] := a[n] = Module[{npr, s}, If[n <= 1, 1, npr = n-2; s = 9*npr*(npr + 3)*a[n-2] - 2*(5*npr^2 + 26*npr + 30)*a[n-1]; -s/(npr + 4)/(npr + 5)]];

%t Table[a[n], {n, 0, 22}] (* _Jean-François Alcover_, Nov 29 2017, after _R. J. Mathar_ *)

%o (PARI) seq(n)={my(a=vector(n+1)); a[1]=1; a[2]=1; for(n=2, n, a[n+1] = (2*(5*n^2 + 6*n - 2)*a[n] - 9*(n - 2)*(n + 1)*a[n-1])/((n + 2)*(n + 3))); a} \\ _Andrew Howroyd_, Dec 26 2019

%K nonn

%O 0,3

%A _N. J. A. Sloane_, Sep 22 2012