This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A289684 #36 Mar 20 2025 11:59:12
%S A289684 1,2,9,42,199,950,4554,21884,105323,507398,2446022,11796884,56912838,
%T A289684 274630876,1325431956,6397576888,30882340531,149084312006,
%U A289684 719736965358,3474807470756,16776410481266,80998307687668,391074406408716,1888199373821896,9116752061308798
%N A289684 Mixing moments for the waiting time in an M/G/1 waiting queue.
%C A289684 In the preprint and the paper by Karpov et al., a(n) (resp. 2*a(n)) is the size of the Condorcet domain on 2*n (resp. 2*n+1) alternatives defined by the so-called even 1N33N1 scheme, cf. A144685. - _Andrey Zabolotskiy_, Jan 27 2024
%H A289684 Robert Israel, <a href="/A289684/b289684.txt">Table of n, a(n) for n = 0..1461</a>
%H A289684 J. Abate and W. Whitt, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL13/Whitt/whitt2.html">Integer Sequences from Queueing Theory</a>, J. Int. Seq. 13 (2010), 10.5.5, eq. (30) and (32).
%H A289684 Alexander Karpov, Klas Markström, Søren Riis and Bei Zhou, <a href="https://arxiv.org/abs/2308.02817">Bipartite peak-pit domains</a>, arXiv:2308.02817 [cs.DM], 2023-2024.
%H A289684 Alexander Karpov, Klas Markström, Søren Riis, and Bei Zhou, <a href="https://doi.org/10.1016/j.dam.2025.03.007">Coherent domains and improved lower bounds for the maximum size of Condorcet domains</a>, Discrete Applied Mathematics, Volume 370, Pages 57-70 (2025). See pp. 68-69.
%F A289684 G.f.: 1/(2-A000108(x)^2), where A000108(x) is the generating function of the Catalan Numbers.
%F A289684 Conjecture: n*a(n) + 2*(-4*n+3)*a(n-1) + 12*(n-2)*a(n-2) + 8*(2*n-3)*a(n-3) = 0.
%F A289684 From _Robert Israel_, Mar 31 2019: (Start)
%F A289684 Conjecture verified (for n >= 4) using the differential equation (16*x^3 + 12*x^2 - 8*x + 1)*y' + (24*x^2 - 2)*y -12*x^2 + 2*x = 0 satisfied by the g.f.
%F A289684 a(n) ~ (sqrt(2)/4)*(2 + 2*sqrt(2))^n. (End)
%p A289684 f:= gfun:-rectoproc({n*a(n) +2*(-4*n+3)*a(n-1) +12*(n-2)*a(n-2) +8*(2*n-3)*a(n-3),a(0)=1,a(1)=2,a(2)=9,a(3)=42},a(n),remember):
%p A289684 map(f, [$0..50]); # _Robert Israel_, Mar 31 2019
%t A289684 CoefficientList[2 x^2/(4 x^2 + 2x + Sqrt[1 - 4x] - 1) + O[x]^25, x] (* _Jean-François Alcover_, Aug 26 2022 *)
%o A289684 (Sage) (2*x^2/(4*x^2+2*x+sqrt(1-4*x)-1)).series(x, 25).coefficients(x, sparse=False) # _Stefano Spezia_, Mar 19 2025
%Y A289684 Cf. A000108, A144685.
%K A289684 nonn,easy
%O A289684 0,2
%A A289684 _R. J. Mathar_, Jul 09 2017