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A187249 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).

%I A187249 #15 Jul 26 2022 13:21:30
%S A187249 0,1,3,11,48,248,1504,10560,84544,761024,7610496,83715968,1004592640,
%T A187249 13059706368,182835893248,2742538406912,43880614526976,
%U A187249 745970446991360,13427468045910016,255121892872421376,5102437857448689664,107151195006423007232,2357326290141307207680
%N A187249 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 A187249 a(n) = Sum_{k>=0} k * A187247(n,k).
%H A187249 Alois P. Heinz, <a href="/A187249/b187249.txt">Table of n, a(n) for n = 0..450</a>
%F A187249 E.g.f.: g(z) = (1/4)[exp(2z) - 1 +2z]/(1-z).
%F A187249 a(n) ~ (exp(2)+1)/4 * n! = 2.09726402473266... * n!. - _Vaclav Kotesovec_, Mar 15 2014
%F A187249 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 A187249 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 A187249 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 A187249 # second Maple program:
%p A187249 b:= proc(n) option remember; expand(
%p A187249       `if`(n=0, 1, add(b(n-j)*binomial(n-1, j-1)*
%p A187249       `if`(j=1, x, (j-1)!+2^(j-2)*(x-1)), j=1..n)))
%p A187249     end:
%p A187249 a:= n-> (p-> add(coeff(p, x, i)*i, i=0..n))(b(n)):
%p A187249 seq(a(n), n=0..30);  # _Alois P. Heinz_, Apr 15 2017
%t A187249 CoefficientList[Series[(E^(2*x)-1+2*x)/(4*(1-x)), {x, 0, 20}], x]* Range[0, 20]! (* _Vaclav Kotesovec_, Mar 15 2014 *)
%Y A187249 Cf. A187247.
%K A187249 nonn
%O A187249 0,3
%A A187249 _Emeric Deutsch_, Mar 07 2011