A187249 - OEIS

%I #15 Jul 26 2022 13:21:30

%S 0,1,3,11,48,248,1504,10560,84544,761024,7610496,83715968,1004592640,

%T 13059706368,182835893248,2742538406912,43880614526976,

%U 745970446991360,13427468045910016,255121892872421376,5102437857448689664,107151195006423007232,2357326290141307207680

%N Number of cycles with at most 2 alternating runs in all permutations of [n] (it is assumed that the smallest element of the cycle is in the first position).

%C a(n) = Sum_{k>=0} k * A187247(n,k).

%H Alois P. Heinz, <a href="/A187249/b187249.txt">Table of n, a(n) for n = 0..450</a>

%F E.g.f.: g(z) = (1/4)[exp(2z) - 1 +2z]/(1-z).

%F a(n) ~ (exp(2)+1)/4 * n! = 2.09726402473266... * n!. - _Vaclav Kotesovec_, Mar 15 2014

%F D-finite with recurrence a(n) +(-n-2)*a(n-1) +2*(n-1)*a(n-2)=0. - _R. J. Mathar_, Jul 26 2022

%e a(3)=11 because in (1)(2)(3), (1)(23), (12)(3), (13)(2), (123), and (132) all cycles have at most  2 alternating runs.

%p g := (1/4*(exp(2*z)-1+2*z))/(1-z): gser := series(g, z = 0, 25): seq(factorial(n)*coeff(gser, z, n), n = 0 .. 22);

%p # second Maple program:

%p b:= proc(n) option remember; expand(

%p       `if`(n=0, 1, add(b(n-j)*binomial(n-1, j-1)*

%p       `if`(j=1, x, (j-1)!+2^(j-2)*(x-1)), j=1..n)))

%p     end:

%p a:= n-> (p-> add(coeff(p, x, i)*i, i=0..n))(b(n)):

%p seq(a(n), n=0..30);  # _Alois P. Heinz_, Apr 15 2017

%t CoefficientList[Series[(E^(2*x)-1+2*x)/(4*(1-x)), {x, 0, 20}], x]* Range[0, 20]! (* _Vaclav Kotesovec_, Mar 15 2014 *)

%Y Cf. A187247.

%K nonn

%O 0,3

%A _Emeric Deutsch_, Mar 07 2011