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A153742 Number of elements in wreath product C_3 wr S_n that alternate up/not-up with respect to a weak product ordering.

%I A153742 #25 Jul 30 2023 19:10:30
%S A153742 3,6,44,201,2436,16768,284388,2610633,56926096,653221506,17409078576,
%T A153742 239721136817,7550440414752,121296879540684,4408222329882272,
%U A153742 80934331054201905,3333529520918540544,68853515512316939422
%N A153742 Number of elements in wreath product C_3 wr S_n that alternate up/not-up with respect to a weak product ordering.
%H A153742 G. C. Greubel, <a href="/A153742/b153742.txt">Table of n, a(n) for n = 1..400</a>
%H A153742 A. Niedermaier and J. Remmel, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL13/Remmel/remmel.html">Analogues of Up-down Permutations for Colored Permutations</a>, J. Int. Seq. 13 (2010), 10.5.6.
%F A153742 E.g.f.: (2 + 2*sin(x) + 4*x*cos(x) - x^2*sin(x))/(2*cos(x) - 4*x*sin(x) -x^2*cos(x)).
%F A153742 a(n)/n! ~ c / r^(n+1) where r = 0.59974142102782394317972557684 is the root of the equation 4*r*tan(r) = (2-r^2), c = 4*sqrt(4 + 12*r^2 + r^4)/(12 + 16*r^2 + r^4) = 1.0837719267197115958973167583838141520381872675225558954477173... if n is even and c = (8 + 24*r^2 + 2*r^4)/(12 + 16*r^2 + r^4) = 1.5747968742391725511892660696837072745667493434277868133205599... if n is odd. - _Vaclav Kotesovec_, Aug 27 2016
%e A153742 Viewing elements in one-line notation as a list of ordered pairs with first entries in [3] and second entries forming a permutation in S_n, two of the 44 up/not-up elements for n=3 are (1,2) (3,3) (1,1) and (1,1) (1,3) (2,2). Note that the first element goes up/down and the second goes up/not-up with respect to the weak product ordering on ordered pairs.
%t A153742 Rest[CoefficientList[Series[(2 + 2*Sin[x] + 4 x*Cos[x] - x^2*Sin[x])/(2*Cos[x] - 4*x*Sin[x] - x^2*Cos[x]), {x, 0, 50}], x]*Range[0, 50]!] (* _G. C. Greubel_, Aug 27 2016 *)
%K A153742 nonn
%O A153742 1,1
%A A153742 _Andrew Niedermaier_, Dec 31 2008