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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.
6
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
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
KEYWORD
nonn
AUTHOR
Emeric Deutsch, Feb 27 2011
STATUS
approved