cp's OEIS Frontend

A186768 Number of nonincreasing odd cycles in all permutations of {1,2,...,n}. 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 A186768 #18 Apr 13 2017 10:56:55
%S A186768 0,0,0,1,4,43,258,2525,20200,222119,2221190,28061889,336742668,
%T A186768 4856656283,67993187962,1107076110629,17713217770064,322047491979087,
%U A186768 5796854855623566,116542615962575753,2330852319251515060,51380800712458456259
%N A186768 Number of nonincreasing odd cycles in all permutations of {1,2,...,n}. 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 A186768 a(n) = Sum(k*A186766(n,k), k>=0).
%H A186768 Vincenzo Librandi, <a href="/A186768/b186768.txt">Table of n, a(n) for n = 0..200</a>
%F A186768 E.g.f.: g(z)=[log((1+z)/(1-z))-2sinh(z)]/(2(1-z)).
%F A186768 a(n) ~ n!/2 * (log(2*n) + gamma - exp(1) + exp(-1)), where gamma is the Euler-Mascheroni constant (A001620). - _Vaclav Kotesovec_, Oct 07 2013
%e A186768 a(3)=1 because in (1)(2)(3), (1)(23), (12)(3), (13)(2), (123), and (132) we have a total of 0+0+0+0+0+1 =1 increasing odd cycles.
%p A186768 g := ((ln((1+z)/(1-z))-2*sinh(z))*1/2)/(1-z): gser := series(g, z = 0, 27): seq(factorial(n)*coeff(gser, z, n), n = 0 .. 21);
%t A186768 CoefficientList[Series[(Log[(1+x)/(1-x)]-2*Sinh[x])/(2*(1-x)), {x, 0, 20}], x]* Range[0, 20]! (* _Vaclav Kotesovec_, Oct 07 2013 *)
%Y A186768 Cf. A186761, A186763, A186764, A186766, A186769, A184958.
%K A186768 nonn
%O A186768 0,5
%A A186768 _Emeric Deutsch_, Feb 27 2011
%E A186768 Typo in e.g.f. corrected by _Vaclav Kotesovec_, Oct 07 2013