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A336598 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.

%I A336598 #35 Aug 12 2020 01:35:08
%S A336598 1,4,2,21,18,6,144,156,96,24,1245,1500,1260,600,120,13140,16470,16560,
%T A336598 11160,4320,720,164745,207270,231210,194040,108360,35280,5040,2399040,
%U A336598 2976120,3507840,3402000,2419200,1149120,322560,40320
%N A336598 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 A336598 Donovan Young, <a href="/A336598/b336598.txt">Table of n, a(n) for n = 1..9870</a> (Rows 1..140).
%H A336598 Donovan Young, <a href="https://arxiv.org/abs/2007.13868">A critical quartet for queuing couples</a>, arXiv:2007.13868 [math.CO], 2020.
%F A336598 T(n,k) = n*T(n-1,k) + n*T(n-1,k-1), with T(n,0) = A233481(n) for n > 0.
%F A336598 E.g.f.: x/sqrt(1 - 2*x)/(1 - x*(1 + y)).
%e A336598 Triangle begins:
%e A336598      1;
%e A336598      4,    2;
%e A336598     21,   18,    6;
%e A336598    144,  156,   96,  24;
%e A336598   1245, 1500, 1260, 600, 120;
%e A336598 ...
%e A336598 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 A336598 CoefficientList[Normal[Series[x/Sqrt[1-2*x]/(1-x(1+y)),{x,0,10}]]/.{x^n_.->x^n*n!},{x,y}]
%o A336598 (PARI)
%o A336598 T(n)={[Vecrev(p) | p<-Vec(serlaplace(x/sqrt(1 - 2*x + O(x^n))/(1 - x*(1 + y))))]}
%o A336598 { my(A=T(8)); for(n=1, #A, print(A[n])) } \\ _Andrew Howroyd_, Jul 29 2020
%Y A336598 Row sums are n*A001147(n) for n > 0.
%Y A336598 First column is A233481(n) for n > 0.
%Y A336598 Leading diagonal is A000142(n) for n > 0.
%Y A336598 Sub-leading diagonal is n*A000142(n) for n > 1.
%Y A336598 Cf. A336599, A336600, A336601.
%K A336598 nonn,tabl
%O A336598 1,2
%A A336598 _Donovan Young_, Jul 29 2020