A186767 Number of permutations of {1,2,...,n} having no nonincreasing odd cycles. A cycle (b(1), b(2), ...) is said to be increasing if, when written with its smallest element in the first position, it satisfies b(1)<b(2)<b(3)<... . A cycle is said to be odd if it has an odd number of entries.

1, 1, 2, 5, 20, 77, 472, 2585, 21968, 157113, 1724064, 15229645, 204738624, 2151199429, 34194201472, 416221515169, 7631627843840, 105565890206193, 2192501224174080, 33962775502534165, 787900686999286784, 13509825183288167869

Offset

0,3

Comments

a(n) = A186766(n,0).

Links

Alois P. Heinz, Table of n, a(n) for n = 0..450

Formula

E.g.f.: g(z) = exp(sinh z)/sqrt(1-z^2).

Example

a(3)=5 because we have (1)(2)(3), (1)(23), (12)(3), (13)(2), and (123).

Maple

g := exp(sinh(z))/sqrt(1-z^2): gser := series(g, z = 0, 27): seq(factorial(n)*coeff(gser, z, n), n = 0 .. 21);
# second Maple program:
a:= proc(n) option remember; `if`(n=0, 1, add(a(n-j)*
      binomial(n-1, j-1)*`if`(j::even, (j-1)!, 1), j=1..n))
    end:
seq(a(n), n=0..25);  # Alois P. Heinz, Apr 13 2017

Mathematica

a[n_] := a[n] = If[n==0, 1, Sum[a[n-j]*Binomial[n-1, j-1]*If[EvenQ[j], (j-1)!, 1], {j, 1, n}]];
Table[a[n], {n, 0, 25}] (* Jean-François Alcover, May 18 2017, after Alois P. Heinz *)

Crossrefs

Cf. A186761, A186762, A186763, A186764, A186765, A186766, A186769, A186770.

Sequence in context: A297350 A027041 A355866 * A009737 A280624 A008983

Adjacent sequences: A186764 A186765 A186766 * A186768 A186769 A186770

Keyword

nonn

Author

Emeric Deutsch, Feb 27 2011

Status

approved