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.: (2 + 2*sin(x) + 4*x*cos(x) - x^2*sin(x))/(2*cos(x) - 4*x*sin(x) -x^2*cos(x)).
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
EXAMPLE
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.
MATHEMATICA
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 *)
CROSSREFS
KEYWORD
nonn
AUTHOR
Andrew Niedermaier, Dec 31 2008
STATUS
approved