A289684 Mixing moments for the waiting time in an M/G/1 waiting queue.

1, 2, 9, 42, 199, 950, 4554, 21884, 105323, 507398, 2446022, 11796884, 56912838, 274630876, 1325431956, 6397576888, 30882340531, 149084312006, 719736965358, 3474807470756, 16776410481266, 80998307687668, 391074406408716, 1888199373821896, 9116752061308798

Offset: 0

Author

R. J. Mathar, Jul 09 2017

Keywords

• nonn
• easy

Comments

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

Links

• Robert Israel, Table of n, a(n) for n = 0..1461
• J. Abate and W. Whitt, Integer Sequences from Queueing Theory, J. Int. Seq. 13 (2010), 10.5.5, eq. (30) and (32).
• Alexander Karpov, Klas Markström, Søren Riis and Bei Zhou, Bipartite peak-pit domains, arXiv:2308.02817 [cs.DM], 2023-2024.
• Alexander Karpov, Klas Markström, Søren Riis, and Bei Zhou, Coherent domains and improved lower bounds for the maximum size of Condorcet domains, Discrete Applied Mathematics, Volume 370, Pages 57-70 (2025). See pp. 68-69.

Crossrefs

Cf. A000108, A144685.

Programs

Maple

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):
map(f, [$0..50]); # Robert Israel, Mar 31 2019

Mathematica

CoefficientList[2 x^2/(4 x^2 + 2x + Sqrt[1 - 4x] - 1) + O[x]^25, x] (* Jean-François Alcover, Aug 26 2022 *)

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

Formula

G.f.: 1/(2-A000108(x)^2), where A000108(x) is the generating function of the Catalan Numbers.

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.

From Robert Israel, Mar 31 2019: (Start)

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.

a(n) ~ (sqrt(2)/4)*(2 + 2*sqrt(2))^n. (End)