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A336601 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 excluded by (i.e., are outside and do not contain) the marked chord.

%I A336601 #24 Oct 21 2020 03:43:48
%S A336601 1,4,2,22,16,7,160,136,88,36,1464,1344,1044,624,249,16224,15504,13344,
%T A336601 9624,5484,2190,211632,206592,188952,152832,104322,58080,23535,
%U A336601 3179520,3139200,2977920,2594880,1990080,1309680,725040,299880,54092160,53729280,52096320,47681280,39652560,29174400,18809640,10473120,4426065
%N A336601 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 excluded by (i.e., are outside and do not contain) the marked chord.
%H A336601 Donovan Young, <a href="/A336601/b336601.txt">Table of n, a(n) for n = 1..9870</a>
%H A336601 Donovan Young, <a href="https://arxiv.org/abs/2007.13868">A critical quartet for queuing couples</a>, arXiv:2007.13868 [math.CO], 2020.
%F A336601 E.g.f.: arctan(x*(1 - y)/sqrt((1 - 2*x)*(1 - 2*x*y)))/(1 - y)/sqrt(1 - 2*x).
%e A336601 Triangle begins:
%e A336601      1;
%e A336601      4,    2;
%e A336601     22,   16,    7;
%e A336601    160,  136,   88,  36;
%e A336601   1464, 1344, 1044, 624, 249;
%e A336601 ...
%e A336601 For n = 2 and k = 1, let the four vertices be {1,2,3,4}. The marked chord can either be (1,2) and it excludes one other chord, namely (3,4), or vice-versa, hence T(2,1) = 2.
%t A336601 CoefficientList[Normal[Series[1/(1-y)/Sqrt[1-2*x]*ArcTan[(x*(1-y))/Sqrt[(1-2*x)]/Sqrt[1-2*y*x]],{x,0,10}]]/.{x^n_.->x^n*n!},{x,y}]
%Y A336601 Row sums are n*A001147(n) for n > 0.
%Y A336601 The first column is A087547(n) for n > 0.
%Y A336601 Leading diagonal is A034430(n-1) for n > 0.
%Y A336601 Cf. A336598, A336599, A336600.
%K A336601 nonn,tabl
%O A336601 1,2
%A A336601 _Donovan Young_, Jul 31 2020