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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) . A cycle is said to be odd if it has an odd number of entries.

%I A186762 #12 May 04 2023 18:56:50
%S A186762 1,0,1,1,9,33,235,1517,12593,111465,1122819,12313409,147949593,
%T A186762 1922353925,26918452691,403744456541,6460109224801,109820584161393,
%U A186762 1976779056442179,37558742545087481,751175283283221129,15774677696321630525,347042934659313999539,7981987292809647817237
%N 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.
%C A186762 a(n) = A186761(n,0).
%H A186762 Alois P. Heinz, <a href="/A186762/b186762.txt">Table of n, a(n) for n = 0..450</a>
%F A186762 E.g.f.: g(z) = exp(-sinh z)/(1-z).
%F A186762 a(n) ~ exp(-sinh(1)) * n! = 0.308756853522... * n!. - _Vaclav Kotesovec_, Mar 16 2014
%e A186762 a(3)=1 because we have (132).
%e A186762 a(4)=9 because we have (12)(34), (13)(24), (14)(23), and the six cyclic permutations of {1,2,3,4}.
%p A186762 g := exp(-sinh(z))/(1-z): gser := series(g, z = 0, 27): seq(factorial(n)*coeff(gser, z, n), n = 0 .. 23);
%p A186762 # second Maple program:
%p A186762 a:= proc(n) option remember; `if`(n=0, 1, add(a(n-j)*
%p A186762      ((j-1)!-irem(j, 2))*binomial(n-1, j-1), j=1..n))
%p A186762     end:
%p A186762 seq(a(n), n=0..23);  # _Alois P. Heinz_, May 04 2023
%t A186762 CoefficientList[Series[E^(-Sinh[x])/(1-x), {x, 0, 20}], x] * Range[0, 20]! (* _Vaclav Kotesovec_, Mar 16 2014 *)
%Y A186762 Cf. A186761, A186763, A186764, A186765, A186766, A186769.
%K A186762 nonn
%O A186762 0,5
%A A186762 _Emeric Deutsch_, Feb 27 2011