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%I A186763 #30 Jul 26 2022 16:36:52
%S A186763 0,1,2,7,28,141,846,5923,47384,426457,4264570,46910271,562923252,
%T A186763 7318002277,102452031878,1536780478171,24588487650736,418004290062513,
%U A186763 7524077221125234,142957467201379447,2859149344027588940,60042136224579367741
%N A186763 Number of increasing odd cycles in all permutations of {1,2,...,n}.
%C A186763 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)<... .
%C A186763 A cycle is said to be odd if it has an odd number of entries.
%H A186763 Seiichi Manyama, <a href="/A186763/b186763.txt">Table of n, a(n) for n = 0..449</a>
%F A186763 a(n) = Sum_{k>=0} k*A186761(n,k).
%F A186763 E.g.f.: (sinh z)/(1-z).
%F A186763 a(n) ~ n! * (exp(2)-1)*exp(-1)/2. - _Vaclav Kotesovec_, Oct 05 2013
%F A186763 a(n) = (exp(1)*Gamma(1+n,1) - exp(-1)*Gamma(1+n,-1))/2 = (A000522(n) - A000166(n))/2. - _Peter Luschny_, Dec 18 2017
%F A186763 a(n) = n! * Sum_{k=0..floor((n-1)/2)} 1 / (2*k+1)!. - _Ilya Gutkovskiy_, Jul 16 2021
%F A186763 D-finite with recurrence a(n) -n*a(n-1) -a(n-2) +(n-2)*a(n-3)=0. - _R. J. Mathar_, Jul 26 2022
%e A186763 a(3)=7 because in (1)(2)(3), (1)(23), (12)(3), (13)(2), (123), and (132) we have a total of 3+1+1+1+1+0=7 increasing odd cycles.
%p A186763 g := sinh(z)/(1-z): gser := series(g, z = 0, 27): seq(factorial(n)*coeff(gser, z, n), n = 0 .. 21);
%p A186763 # Alternatively:
%p A186763 A186763 := n -> (exp(1)*GAMMA(1+n,1) - exp(-1)*GAMMA(1+n,-1))/2:
%p A186763 seq(simplify(A186763(n)), n=0..21); # _Peter Luschny_, Dec 18 2017
%t A186763 a=0;Table[a=n*a+(1/2)(1-(-1)^n),{n,0,30}] (* _Vladimir Joseph Stephan Orlovsky_, Apr 02 2011 *)
%t A186763 CoefficientList[Series[Sinh[x]/(1-x), {x, 0, 20}], x]* Range[0, 20]! (* _Vaclav Kotesovec_, Oct 05 2013 *)
%Y A186763 Cf. A000166, A000522, A009628, A186761, A186762, A186764, A186765, A186766, A186768, A186769, A184958.
%K A186763 nonn
%O A186763 0,3
%A A186763 _Emeric Deutsch_, Feb 27 2011

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