cp's OEIS Frontend

A186367 Number of cycles in all cycle-up-down permutations of {1,2,...,n}. A permutation is said to be cycle-up-down if it is a product of up-down cycles. 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(3)<... .

%I A186367 #24 Aug 30 2018 08:39:25
%S A186367 1,3,10,38,165,812,4478,27408,184529,1356256,10809786,92892928,
%T A186367 856329253,8430600960,88292571934,980197173248,11499036105537,
%U A186367 142147625652224,1846872283846922,25161923756064768,358706981125488581,5340498034862030848
%N A186367 Number of cycles in all cycle-up-down permutations of {1,2,...,n}. A permutation is said to be cycle-up-down if it is a product of up-down cycles. 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 A186367 Vincenzo Librandi, <a href="/A186367/b186367.txt">Table of n, a(n) for n = 1..200</a>
%H A186367 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 A186367 E.g.f.: -log(1-sin(z)) / (1-sin(z)).
%F A186367 a(n) = Sum_{k=1..n} k*A186366(n,k).
%F A186367 a(n) ~ n!*n*2^(n+3)/Pi^(n+2)*(2*log(n/Pi) + 2*gamma + 3*log(2) - 2), where gamma is the Euler-Mascheroni constant (A001620). - _Vaclav Kotesovec_, Oct 02 2013
%e A186367 a(3) = 10 because the cycle-up-down permutations (1)(2)(3), (12)(3), (13)(2), (1)(23), and (132), have a total of 3+2+2+2+1=10 cycles.
%p A186367 g := -ln(1-sin(z))/(1-sin(z)): gser := series(g, z = 0, 25): seq(factorial(n)*coeff(gser, z, n), n = 1 .. 22);
%t A186367 Rest[CoefficientList[Series[-Log[1-Sin[x]]/(1-Sin[x]), {x, 0, 20}], x]* Range[0, 20]!] (* _Vaclav Kotesovec_, Oct 02 2013 *)
%o A186367 (PARI) x='x+O('x^30); Vec(serlaplace(-log(1-sin(x))/(1-sin(x)))) \\ _G. C. Greubel_, Aug 30 2018
%Y A186367 Cf. A186366.
%K A186367 nonn
%O A186367 1,2
%A A186367 _Emeric Deutsch_, Feb 28 2011