A336601 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 excluded by (i.e., are outside and do not contain) the marked chord.

1, 4, 2, 22, 16, 7, 160, 136, 88, 36, 1464, 1344, 1044, 624, 249, 16224, 15504, 13344, 9624, 5484, 2190, 211632, 206592, 188952, 152832, 104322, 58080, 23535, 3179520, 3139200, 2977920, 2594880, 1990080, 1309680, 725040, 299880, 54092160, 53729280, 52096320, 47681280, 39652560, 29174400, 18809640, 10473120, 4426065

Offset: 1

Author

Donovan Young, Jul 31 2020

Keywords

• nonn
• tabl

Examples

			Triangle begins:
     1;
     4,    2;
    22,   16,    7;
   160,  136,   88,  36;
  1464, 1344, 1044, 624, 249;
...
For n = 2 and k = 1, let the four vertices be {1,2,3,4}. The marked chord can either be (1,2) and it excludes one other chord, namely (3,4), or vice-versa, hence T(2,1) = 2.
		

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.

The first column is A087547(n) for n > 0.

Leading diagonal is A034430(n-1) for n > 0.

Cf. A336598, A336599, A336600.

Programs

Mathematica

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

Formula

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