A153741 Number of elements in wreath product C_2 wr S_n that alternate up/not-up with respect to a weak product ordering.

2, 3, 14, 49, 376, 1987, 21328, 150337, 2074624, 18279971, 308317184, 3259985969, 64981320704, 801591982115, 18436312819712, 259914703640065, 6774998673915904, 107452993132016323, 3130412454801965056

Offset: 1

Author

Andrew Niedermaier, Dec 31 2008

Keywords

• nonn

Examples

			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.
		

Links

• G. C. Greubel, Table of n, a(n) for n = 1..400
• A. Niedermaier and J. Remmel, Analogues of Up-down Permutations for Colored Permutations, J. Int. Seq. 13 (2010), 10.5.6.

Crossrefs

Cf. A069855.

Programs

Mathematica

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 *)

Formula

E.g.f.: (1 + sin(x) + x*cos(x))/(cos(x) - x*sin(x)).

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