A336600 Triangle read by rows: T(n,k) is the number of linear chord diagrams on 2n vertices with one marked chord such that exactly k of the remaining n-1 chords contain the marked chord.

1, 5, 1, 32, 11, 2, 260, 116, 38, 6, 2589, 1344, 594, 174, 24, 30669, 17529, 9294, 3774, 984, 120, 422232, 257487, 153852, 76782, 28272, 6600, 720, 6633360, 4234320, 2746260, 1576980, 726480, 242640, 51120, 5040, 117193185, 77358600, 53170380, 33718500, 18171360, 7693200, 2340720, 448560, 40320

Offset: 1

Author

Donovan Young, Jul 30 2020

Keywords

• nonn
• tabl

Examples

			Triangle begins:
     1;
     5,    1;
    32,   11,    2;
   260,  116,   38,   6;
  2589, 1344,  594, 174, 24;
...
For n = 2 and k = 1, let the four vertices be {1,2,3,4}. The marked chord can only be (2,3) and it is contained by one other chord, namely (1,4), hence T(2,1) = 1.
		

Links

• Donovan Young, Table of n, a(n) for n = 1..9870
• Donovan Young, A critical quartet for queuing couples, arXiv:2007.13868 [math.CO], 2020.

Crossrefs

Row sums are n*A001147(n) for n > 0.

Leading diagonal is A000142(n-1) for n > 0.

Sub-leading diagonal is A001344(n-2) for n > 1.

Cf. A336598, A336599, A336601.

Programs

Mathematica

CoefficientList[Normal[Series[Log[(1-x*(1+y))/(1-2*x)]/(1-y)/Sqrt[1-2*x],{x,0,10}]]/.{x^n_.->x^n*n!},{x,y}]

Formula

E.g.f.: log((1 - x*(1 + y))/(1 - 2*x))/(1 - y)/sqrt(1 - 2*x).