A186359 Number of permutations of {1,2,...,n} having no 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)<... .

1, 0, 0, 1, 4, 19, 114, 799, 6392, 57527, 575270, 6327971, 75935652, 987163475, 13820288650, 207304329751, 3316869276016, 56386777692271, 1014961998460878, 19284277970756683, 385685559415133660, 8099396747717806859, 178186728449791750898

Offset

0,5

Comments

a(n)=A186358(n,0).

Links

Vincenzo Librandi, Table of n, a(n) for n = 0..200

E. Deutsch and S. Elizalde, Cycle up-down permutations, arXiv:0909.5199v1 [math.CO].

Formula

E.g.f.=(1-sin z)/(1-z).

a(2m-1)=(2m-1)!*Sum((-1)^j/(2j-1)!, j=2..m).

a(2m)=(2m)!*Sum((-1)^j/(2j-1)!, j=2..m).

a(n) ~ n! * (1-sin(1)). - Vaclav Kotesovec, Oct 02 2013

Example

a(4)=4 because we have (1432), (1342), (1243), and (1234).

Maple

g := (1-sin(z))/(1-z): gser := series(g, z = 0, 25): seq(factorial(n)*coeff(gser, z, n), n = 0 .. 22);

Mathematica

CoefficientList[Series[(1-Sin[x])/(1-x), {x, 0, 20}], x]* Range[0, 20]! (* Vaclav Kotesovec, Oct 02 2013 *)

Crossrefs

Cf. A186358

Sequence in context: A122835 A013185 A353607 * A203268 A209673 A261497

Adjacent sequences: A186356 A186357 A186358 * A186360 A186361 A186362

Keyword

nonn

Author

Emeric Deutsch, Feb 28 2011

Status

approved