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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.
6

%I #20 Aug 11 2020 16:43:31

%S 1,5,1,33,9,3,279,87,39,15,2895,975,495,255,105,35685,12645,6885,4005,

%T 2205,945,509985,187425,106785,66465,41265,23625,10395,8294895,

%U 3133935,1843695,1198575,795375,513135,301455,135135,151335135,58437855,35213535,23601375,16343775,11263455,7453215,4459455,2027025

%N 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.

%H Donovan Young, <a href="/A336599/b336599.txt">Table of n, a(n) for n = 1..9870</a>

%H Donovan Young, <a href="https://arxiv.org/abs/2007.13868">A critical quartet for queuing couples</a>, arXiv:2007.13868 [math.CO], 2020.

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

%e Triangle begins:

%e 1;

%e 5, 1;

%e 33, 9, 3;

%e 279, 87, 39, 15;

%e 2895, 975, 495, 255, 105;

%e ...

%e 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.

%t 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}]

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

%Y Leading diagonal is A001147(n-1) for n > 0.

%Y The first column is A129890(n-1) for n > 0.

%Y The second column is A035101(n+1) for n > 0.

%Y Cf. A336598, A336600, A336601.

%K nonn,tabl

%O 1,2

%A _Donovan Young_, Jul 29 2020