A186362 - OEIS

%I #21 Aug 30 2018 08:39:33

%S 0,0,0,1,8,59,458,3865,35688,360127,3956214,47096633,604722604,

%T 8337692687,122932350162,1930964182649,32201197533072,568323756438079,

%U 10585305178638446,207520767220183033,4272031355927379828,92145190111463378863,2078280173125850650890

%N Number of non-up-down cycles in all permutations of {1,2,...,n}. A cycle (b(1), b(2), ...) is said to be up-down if, when written with its smallest element in the first position, it satisfies b(1)<b(2)>b(3)<... .

%H Vincenzo Librandi, <a href="/A186362/b186362.txt">Table of n, a(n) for n = 0..200</a>

%H E. Deutsch and S. Elizalde, <a href="http://arxiv.org/abs/0909.5199">Cycle up-down permutations</a>, arXiv:0909.5199 [math.CO], 2009.

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

%F a(n) = Sum(k*A186361(n,k), k>=0).

%F a(n) = n!*Sum(((j-1)!-E(j-1))/j!, j=1..n), where E(i)=A000111(i) are the Euler (or up-down) numbers.

%F a(n) ~ n! * (log(n) + gamma + log(1-sin(1))), where gamma is the Euler-Mascheroni constant (A001620). - _Vaclav Kotesovec_, Oct 02 2013

%e a(3) = 10 because the permutations (1)(2)(3), (12)(3), (13)(2), (1)(23), (132), and (123) have a total of 3 + 2 + 2 + 2 + 0 + 1 = 10 up-down cycles.

%p g := ln((1-sin(z))/(1-z))/(1-z): gser := series(g, z = 0, 25): seq(factorial(n)*coeff(gser, z, n), n = 0 .. 22);

%t CoefficientList[Series[Log[(1-Sin[x])/(1-x)]/(1-x), {x, 0, 20}], x]* Range[0, 20]! (* _Vaclav Kotesovec_, Oct 02 2013 *)

%o (PARI) x='x+O('x^30); concat(vector(3), Vec(serlaplace(log((1-sin(x))/(1-x))/(1-x)))) \\ _G. C. Greubel_, Aug 30 2018

%Y Cf. A000111, A186361, A186358, A186360.

%K nonn

%O 0,5

%A _Emeric Deutsch_, Feb 28 2011