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(3)<... .

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

Offset: 0

Author

Emeric Deutsch, Feb 28 2011

Keywords

• nonn

Comments

a(n) = A186363(n,0).

Hankel transform is A154604. Binomial transform is A000111(n+1). - Paul Barry, Apr 11 2011

Examples

			a(4) = 5 because we have (12)(34),(13)(24),(1324),(1423), and (14)(23).
		

Links

• Nathaniel Johnston, Table of n, a(n) for n = 0..198
• Emeric Deutsch and Sergi Elizalde, Cycle up-down permutations, arXiv:0909.5199 [math.CO], 2009; and also, Australas. J. Combin. 50 (2011), 187-199.

Crossrefs

Cf. A186363.

Programs

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 *)

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