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

1,2

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.

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)).

Example

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.

Mathematica

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

Prog

(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

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.

Sequence in context: A157407 A100400 A285439 * A336601 A241437 A030211

Adjacent sequences: A336595 A336596 A336597 * A336599 A336600 A336601

Keyword

nonn,tabl

Author

Donovan Young, Jul 29 2020

Status

approved