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A289684 Mixing moments for the waiting time in a M/G/1 waiting queue. 1
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

LINKS

Robert Israel, Table of n, a(n) for n = 0..1461

J. Abate, W. Whitt, Integer Sequences from Queueing Theory, J. Int. Seq. 13 (2010), 10.5.5, eq. (30) and (32).

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

CROSSREFS

Cf. A000108.

Sequence in context: A330016 A056845 A162273 * A280955 A276508 A347996

Adjacent sequences:  A289681 A289682 A289683 * A289685 A289686 A289687

KEYWORD

nonn,easy

AUTHOR

R. J. Mathar, Jul 09 2017

STATUS

approved

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Last modified January 27 05:50 EST 2022. Contains 350601 sequences. (Running on oeis4.)