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)
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
Keywords
Examples
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}.
Links
- Alois P. Heinz, Table of n, a(n) for n = 0..450
Programs
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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 *)
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
Comments