cp's OEIS Frontend

A336599 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.

%I A336599 #20 Aug 11 2020 16:43:31
%S A336599 1,5,1,33,9,3,279,87,39,15,2895,975,495,255,105,35685,12645,6885,4005,
%T A336599 2205,945,509985,187425,106785,66465,41265,23625,10395,8294895,
%U A336599 3133935,1843695,1198575,795375,513135,301455,135135,151335135,58437855,35213535,23601375,16343775,11263455,7453215,4459455,2027025
%N A336599 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 A336599 Donovan Young, <a href="/A336599/b336599.txt">Table of n, a(n) for n = 1..9870</a>
%H A336599 Donovan Young, <a href="https://arxiv.org/abs/2007.13868">A critical quartet for queuing couples</a>, arXiv:2007.13868 [math.CO], 2020.
%F A336599 E.g.f.: (sqrt(1 - 2*y*x) - sqrt(1 - 2*x))/(1 - 2*x)/(1 - y).
%e A336599 Triangle begins:
%e A336599      1;
%e A336599      5,    1;
%e A336599     33,    9,    3;
%e A336599    279,   87,   39,  15;
%e A336599   2895,  975,  495, 255, 105;
%e A336599 ...
%e A336599 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 A336599 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 A336599 Row sums are n*A001147(n) for n > 0.
%Y A336599 Leading diagonal is A001147(n-1) for n > 0.
%Y A336599 The first column is A129890(n-1) for n > 0.
%Y A336599 The second column is A035101(n+1) for n > 0.
%Y A336599 Cf. A336598, A336600, A336601.
%K A336599 nonn,tabl
%O A336599 1,2
%A A336599 _Donovan Young_, Jul 29 2020