|
%I #28 Jan 28 2024 01:40:37
%S 1,2,9,42,199,950,4554,21884,105323,507398,2446022,11796884,56912838,
%T 274630876,1325431956,6397576888,30882340531,149084312006,
%U 719736965358,3474807470756,16776410481266,80998307687668,391074406408716,1888199373821896,9116752061308798
%N Mixing moments for the waiting time in an M/G/1 waiting queue.
%C In 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 Robert Israel, <a href="/A289684/b289684.txt">Table of n, a(n) for n = 0..1461</a>
%H 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 Alexander Karpov, Klas Markström, Søren Riis and Bei Zhou, <a href="https://arxiv.org/abs/2308.02817">Set-alternating schemes: A new class of large Condorcet domains</a>, arXiv:2308.02817 [cs.DM], 2023.
%F G.f.: 1/(2-A000108(x)^2), where A000108(x) is the generating function of the Catalan Numbers.
%F 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 From _Robert Israel_, Mar 31 2019: (Start)
%F 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 a(n) ~ (sqrt(2)/4)*(2 + 2*sqrt(2))^n. (End)
%p 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 map(f, [$0..50]); # _Robert Israel_, Mar 31 2019
%t CoefficientList[2 x^2/(4 x^2 + 2x + Sqrt[1 - 4x] - 1) + O[x]^25, x] (* _Jean-François Alcover_, Aug 26 2022 *)
%Y Cf. A000108, A144685.
%K nonn,easy
%O 0,2
%A _R. J. Mathar_, Jul 09 2017
|
