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%I #35 Aug 12 2020 01:35:08
%S 1,4,2,21,18,6,144,156,96,24,1245,1500,1260,600,120,13140,16470,16560,
%T 11160,4320,720,164745,207270,231210,194040,108360,35280,5040,2399040,
%U 2976120,3507840,3402000,2419200,1149120,322560,40320
%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 cross the marked chord.
%H Donovan Young, <a href="/A336598/b336598.txt">Table of n, a(n) for n = 1..9870</a> (Rows 1..140).
%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 T(n,k) = n*T(n-1,k) + n*T(n-1,k-1), with T(n,0) = A233481(n) for n > 0.
%F E.g.f.: x/sqrt(1 - 2*x)/(1 - x*(1 + y)).
%e Triangle begins:
%e 1;
%e 4, 2;
%e 21, 18, 6;
%e 144, 156, 96, 24;
%e 1245, 1500, 1260, 600, 120;
%e ...
%e For n = 2 and k = 1, let the four vertices be {1,2,3,4}. The marked chord can be either (1,3), and so crossed once by (2,4), or (2,4), and so crossed once by (1,3). Hence T(2,1) = 2.
%t CoefficientList[Normal[Series[x/Sqrt[1-2*x]/(1-x(1+y)),{x,0,10}]]/.{x^n_.->x^n*n!},{x,y}]
%o (PARI)
%o T(n)={[Vecrev(p) | p<-Vec(serlaplace(x/sqrt(1 - 2*x + O(x^n))/(1 - x*(1 + y))))]}
%o { my(A=T(8)); for(n=1, #A, print(A[n])) } \\ _Andrew Howroyd_, Jul 29 2020
%Y Row sums are n*A001147(n) for n > 0.
%Y First column is A233481(n) for n > 0.
%Y Leading diagonal is A000142(n) for n > 0.
%Y Sub-leading diagonal is n*A000142(n) for n > 1.
%Y Cf. A336599, A336600, A336601.
%K nonn,tabl
%O 1,2
%A _Donovan Young_, Jul 29 2020
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