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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.

%I A336600 #22 Jun 20 2025 08:12:19
%S A336600 1,5,1,32,11,2,260,116,38,6,2589,1344,594,174,24,30669,17529,9294,
%T A336600 3774,984,120,422232,257487,153852,76782,28272,6600,720,6633360,
%U A336600 4234320,2746260,1576980,726480,242640,51120,5040,117193185,77358600,53170380,33718500,18171360,7693200,2340720,448560,40320
%N 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.
%H A336600 Donovan Young, <a href="/A336600/b336600.txt">Table of n, a(n) for n = 1..9870</a>
%H A336600 Donovan Young, <a href="https://arxiv.org/abs/2007.13868">A critical quartet for queuing couples</a>, arXiv:2007.13868 [math.CO], 2020.
%F A336600 E.g.f.: log((1 - x*(1 + y))/(1 - 2*x))/(1 - y)/sqrt(1 - 2*x).
%e A336600 Triangle begins:
%e A336600      1;
%e A336600      5,    1;
%e A336600     32,   11,    2;
%e A336600    260,  116,   38,   6;
%e A336600   2589, 1344,  594, 174, 24;
%e A336600 ...
%e A336600 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.
%t A336600 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}]
%Y A336600 Row sums are n*A001147(n) for n > 0.
%Y A336600 Leading diagonal is A000142(n-1) for n > 0.
%Y A336600 Sub-leading diagonal is A001344(n-2) for n > 1.
%Y A336600 Cf. A336598, A336599, A336601.
%K A336600 nonn,tabl
%O A336600 1,2
%A A336600 _Donovan Young_, Jul 30 2020