A336599 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 are contained within the marked chord.

1, 5, 1, 33, 9, 3, 279, 87, 39, 15, 2895, 975, 495, 255, 105, 35685, 12645, 6885, 4005, 2205, 945, 509985, 187425, 106785, 66465, 41265, 23625, 10395, 8294895, 3133935, 1843695, 1198575, 795375, 513135, 301455, 135135, 151335135, 58437855, 35213535, 23601375, 16343775, 11263455, 7453215, 4459455, 2027025

Offset: 1

Author

Donovan Young, Jul 29 2020

Keywords

• nonn
• tabl

Examples

			Triangle begins:
     1;
     5,    1;
    33,    9,    3;
   279,   87,   39,  15;
  2895,  975,  495, 255, 105;
...
For n = 2 and k = 1, let the four vertices be {1,2,3,4}. The marked chord can only be (1,4) and it contains one other chord, namely (2,3), 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 A001147(n-1) for n > 0.

The first column is A129890(n-1) for n > 0.

The second column is A035101(n+1) for n > 0.

Cf. A336598, A336600, A336601.

Programs

Mathematica

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

Formula

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