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%I #24 Jul 30 2023 18:22:59
%S 2,3,14,49,376,1987,21328,150337,2074624,18279971,308317184,
%T 3259985969,64981320704,801591982115,18436312819712,259914703640065,
%U 6774998673915904,107452993132016323,3130412454801965056
%N Number of elements in wreath product C_2 wr S_n that alternate up/not-up with respect to a weak product ordering.
%H G. C. Greubel, <a href="/A153741/b153741.txt">Table of n, a(n) for n = 1..400</a>
%H 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 E.g.f.: (1 + sin(x) + x*cos(x))/(cos(x) - x*sin(x)).
%F a(n) ~ c * n! / r^(n+1), where r = 0.860333589... (=A069855) is the root of the equation sin(r)*r = cos(r), and c = 2/((2+r^2)*sin(r)) = 0.9628268573779... if n is even and c = 2-2/(r^2+2*r*tan(r)) = 1.2701193119933... if n is odd. - _Vaclav Kotesovec_, Sep 25 2013
%e Viewing elements in one-line notation as a list of ordered pairs with first entries in [2] and second entries forming a permutation in S_n, two of the 6 up/not-up elements for n=3 are (1,2) (2,3) (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 Rest[CoefficientList[Series[(1+Sin[x]+x*Cos[x])/(Cos[x]-x*Sin[x]), {x, 0, 20}], x]* Range[0, 20]!] (* _Vaclav Kotesovec_, Sep 25 2013 *)
%Y Cf. A069855.
%K nonn
%O 1,1
%A _Andrew Niedermaier_, Dec 31 2008
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