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

1,2

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.

Formula

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

Example

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.

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}]

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.

Sequence in context: A296043 A377058 A336600 * A066833 A334887 A039813

Adjacent sequences: A336596 A336597 A336598 * A336600 A336601 A336602

Keyword

nonn,tabl

Author

Donovan Young, Jul 29 2020

Status

approved