|
| |
|
|
A186364
|
|
Number of cycle-up-down permutations of {1,2,...,n} having no fixed points. A permutation is said to be cycle-up-down if it is a product of up-down cycles. A cycle (b(1), b(2), ...) is said to be up-down if, when written with its smallest element in the first position, it satisfies b(1)<b(2)>b(3)<... .
|
|
4
|
|
|
|
1, 0, 1, 1, 5, 15, 71, 341, 1945, 12135, 84091, 635281, 5212085, 46091955, 437198711, 4426839821, 47657861425, 543551916975, 6546911178931, 83039587809961, 1106307936885965, 15445529882517195, 225502102290364751, 3436240674908121701, 54555087491802061705
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
|
OFFSET
|
0,5
|
|
|
COMMENTS
|
|
|
|
LINKS
|
|
|
|
FORMULA
|
E.g.f.: exp(-z)/(1-sin(z)).
G.f.: 1/(1-x^2/(1-x-3*x^2/(1-2*x-6*x^2/(1-3*x-10*x^2/(1-.../(1-n*x-((n+1)*(n+2)/2)*x^2/(1-... (continued fraction). - Paul Barry, Apr 11 2011
a(n) ~ n! * n * exp(-Pi/2) * 2^(n+3) / Pi^(n+2). - Vaclav Kotesovec, Oct 08 2013
G.f.: conjecture: T(0), where T(k) = 1 - x^2*(k+1)*(k+2)/(x^2*(k+1)*(k+2) - 2*(1-x*k)*(1-x*(k+1))/T(k+1) ); (continued fraction). - Sergei N. Gladkovskii, Nov 21 2013
|
|
|
EXAMPLE
|
a(4) = 5 because we have (12)(34),(13)(24),(1324),(1423), and (14)(23).
|
|
|
MAPLE
|
g := exp(-z)/(1-sin(z)): gser := series(g, z = 0, 28): seq(factorial(n)*coeff(gser, z, n), n = 0 .. 24);
|
|
|
MATHEMATICA
|
CoefficientList[Series[E^(-x)/(1-Sin[x]), {x, 0, 20}], x]* Range[0, 20]! (* Vaclav Kotesovec, Oct 08 2013 *)
|
|
|
CROSSREFS
|
|
|
|
KEYWORD
|
nonn
|
|
|
AUTHOR
|
|
|
|
STATUS
|
approved
|
| |
|
|
