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

1,1

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.

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

Example

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.

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 *)

Crossrefs

Cf. A069855.

Sequence in context: A128849 A294495 A188289 * A070207 A268559 A346057

Adjacent sequences: A153738 A153739 A153740 * A153742 A153743 A153744

Keyword

nonn

Author

Andrew Niedermaier, Dec 31 2008

Status

approved