A336598 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 cross the marked chord.

1, 4, 2, 21, 18, 6, 144, 156, 96, 24, 1245, 1500, 1260, 600, 120, 13140, 16470, 16560, 11160, 4320, 720, 164745, 207270, 231210, 194040, 108360, 35280, 5040, 2399040, 2976120, 3507840, 3402000, 2419200, 1149120, 322560, 40320

Offset: 1

Author

Donovan Young, Jul 29 2020

Keywords

• nonn
• tabl

Examples

			Triangle begins:
     1;
     4,    2;
    21,   18,    6;
   144,  156,   96,  24;
  1245, 1500, 1260, 600, 120;
...
For n = 2 and k = 1, let the four vertices be {1,2,3,4}. The marked chord can be either (1,3), and so crossed once by (2,4), or (2,4), and so crossed once by (1,3). Hence T(2,1) = 2.
		

Links

• Donovan Young, Table of n, a(n) for n = 1..9870 (Rows 1..140).
• Donovan Young, A critical quartet for queuing couples, arXiv:2007.13868 [math.CO], 2020.

Crossrefs

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

First column is A233481(n) for n > 0.

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

Sub-leading diagonal is n*A000142(n) for n > 1.

Cf. A336599, A336600, A336601.

Programs

Mathematica

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

PARI

T(n)={[Vecrev(p) | p<-Vec(serlaplace(x/sqrt(1 - 2*x + O(x^n))/(1 - x*(1 + y))))]}
{ my(A=T(8)); for(n=1, #A, print(A[n])) } \\ Andrew Howroyd, Jul 29 2020

Formula

T(n,k) = n*T(n-1,k) + n*T(n-1,k-1), with T(n,0) = A233481(n) for n > 0.

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