%I #30 Jul 26 2022 16:36:52

%S 0,1,2,7,28,141,846,5923,47384,426457,4264570,46910271,562923252,

%T 7318002277,102452031878,1536780478171,24588487650736,418004290062513,

%U 7524077221125234,142957467201379447,2859149344027588940,60042136224579367741

%N Number of increasing odd cycles in all permutations of {1,2,...,n}.

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

%H Seiichi Manyama, <a href="/A186763/b186763.txt">Table of n, a(n) for n = 0..449</a>

%F a(n) = Sum_{k>=0} k*A186761(n,k).

%F E.g.f.: (sinh z)/(1-z).

%F a(n) ~ n! * (exp(2)-1)*exp(-1)/2. - _Vaclav Kotesovec_, Oct 05 2013

%F 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 a(n) = n! * Sum_{k=0..floor((n-1)/2)} 1 / (2*k+1)!. - _Ilya Gutkovskiy_, Jul 16 2021

%F 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 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 g := sinh(z)/(1-z): gser := series(g, z = 0, 27): seq(factorial(n)*coeff(gser, z, n), n = 0 .. 21);

%p # Alternatively:

%p A186763 := n -> (exp(1)*GAMMA(1+n,1) - exp(-1)*GAMMA(1+n,-1))/2:

%p seq(simplify(A186763(n)), n=0..21); # _Peter Luschny_, Dec 18 2017

%t a=0;Table[a=n*a+(1/2)(1-(-1)^n),{n,0,30}] (* _Vladimir Joseph Stephan Orlovsky_, Apr 02 2011 *)

%t CoefficientList[Series[Sinh[x]/(1-x), {x, 0, 20}], x]* Range[0, 20]! (* _Vaclav Kotesovec_, Oct 05 2013 *)

%Y Cf. A000166, A000522, A009628, A186761, A186762, A186764, A186765, A186766, A186768, A186769, A184958.

%K nonn

%O 0,3

%A _Emeric Deutsch_, Feb 27 2011