A186762 Number of permutations of {1,2,...,n} having no increasing 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, 0, 1, 1, 9, 33, 235, 1517, 12593, 111465, 1122819, 12313409, 147949593, 1922353925, 26918452691, 403744456541, 6460109224801, 109820584161393, 1976779056442179, 37558742545087481, 751175283283221129, 15774677696321630525, 347042934659313999539, 7981987292809647817237

Offset

0,5

Comments

a(n) = A186761(n,0).

Links

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

Formula

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

a(n) ~ exp(-sinh(1)) * n! = 0.308756853522... * n!. - Vaclav Kotesovec, Mar 16 2014

Example

a(3)=1 because we have (132).
a(4)=9 because we have (12)(34), (13)(24), (14)(23), and the six cyclic permutations of {1,2,3,4}.

Maple

g := exp(-sinh(z))/(1-z): gser := series(g, z = 0, 27): seq(factorial(n)*coeff(gser, z, n), n = 0 .. 23);
# second Maple program:
a:= proc(n) option remember; `if`(n=0, 1, add(a(n-j)*
     ((j-1)!-irem(j, 2))*binomial(n-1, j-1), j=1..n))
    end:
seq(a(n), n=0..23);  # Alois P. Heinz, May 04 2023

Mathematica

CoefficientList[Series[E^(-Sinh[x])/(1-x), {x, 0, 20}], x] * Range[0, 20]! (* Vaclav Kotesovec, Mar 16 2014 *)

Crossrefs

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

Sequence in context: A146445 A046997 A013488 * A343843 A063423 A183939

Adjacent sequences: A186759 A186760 A186761 * A186763 A186764 A186765

Keyword

nonn

Author

Emeric Deutsch, Feb 27 2011

Status

approved