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

0,2

Comments

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

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, Set-alternating schemes: A new class of large Condorcet domains, arXiv:2308.02817 [cs.DM], 2023.

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)

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 *)

Crossrefs

Cf. A000108, A144685.

Sequence in context: A056845 A354302 A162273 * A368764 A280955 A276508

Adjacent sequences: A289681 A289682 A289683 * A289685 A289686 A289687

Keyword

nonn,easy

Author

R. J. Mathar, Jul 09 2017

Status

approved