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A336600
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 contain the marked chord.
5
1, 5, 1, 32, 11, 2, 260, 116, 38, 6, 2589, 1344, 594, 174, 24, 30669, 17529, 9294, 3774, 984, 120, 422232, 257487, 153852, 76782, 28272, 6600, 720, 6633360, 4234320, 2746260, 1576980, 726480, 242640, 51120, 5040, 117193185, 77358600, 53170380, 33718500, 18171360, 7693200, 2340720, 448560, 40320
OFFSET
1,2
LINKS
Donovan Young, A critical quartet for queuing couples, arXiv:2007.13868 [math.CO], 2020.
FORMULA
E.g.f.: log((1 - x*(1 + y))/(1 - 2*x))/(1 - y)/sqrt(1 - 2*x).
EXAMPLE
Triangle begins:
1;
5, 1;
32, 11, 2;
260, 116, 38, 6;
2589, 1344, 594, 174, 24;
...
For n = 2 and k = 1, let the four vertices be {1,2,3,4}. The marked chord can only be (2,3) and it is contained by one other chord, namely (1,4), hence T(2,1) = 1.
MATHEMATICA
CoefficientList[Normal[Series[ CoefficientList[Normal[Series[Log[(1-x*(1+y))/(1-2*x)]/(1-y)/Sqrt[1-2*x], {x, 0, 10}]]/.{x^n_.->x^n*n!}, {x, y}]
CROSSREFS
Row sums are n*A001147(n) for n > 0.
Leading diagonal is A000142(n-1) for n > 0.
Sub-leading diagonal is A001344(n-2) for n > 1.
Sequence in context: A197654 A296043 A377058 * A336599 A066833 A334887
KEYWORD
nonn,tabl
AUTHOR
Donovan Young, Jul 30 2020
STATUS
approved