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A187248 Number of permutations of [n] having only cycles with at least 3 alternating runs (it is assumed that the smallest element of the cycle is in the first position).

%I A187248 #15 Jun 16 2018 13:35:21
%S A187248 1,0,0,0,2,16,104,688,5116,44224,438560,4851136,58603544,764606016,
%T A187248 10715043104,160692920256,2570016145680,43678554864128,
%U A187248 786135111482112,14936420654180864,298733557232591136,6273502048592506112,138018105454095739008,3174423293668325353216
%N A187248 Number of permutations of [n] having only cycles with at least 3 alternating runs (it is assumed that the smallest element of the cycle is in the first position).
%C A187248 a(n) = A187247(n,0).
%H A187248 Alois P. Heinz, <a href="/A187248/b187248.txt">Table of n, a(n) for n = 0..450</a>
%F A187248 E.g.f.: g(z) = exp[(1-2z-exp(2z))/4]/(1-z).
%F A187248 a(n) ~ exp((-1-exp(2))/4) * n!. - _Vaclav Kotesovec_, Mar 18 2014
%e A187248 a(4)=2 because we have (1423) and (1324).
%p A187248 g := exp((1-2*z-exp(2*z))*1/4)/(1-z): gser := series(g, z = 0, 25): seq(factorial(n)*coeff(gser, z, n), n = 0 .. 23);
%p A187248 # second Maple program:
%p A187248 a:= proc(n) option remember;
%p A187248       `if`(n=0, 1, add(a(n-j)*binomial(n-1, j-1)*
%p A187248       `if`(j=1, 0, (j-1)!-2^(j-2)), j=1..n))
%p A187248     end:
%p A187248 seq(a(n), n=0..30);  # _Alois P. Heinz_, Apr 15 2017
%t A187248 a[n_] := a[n] = If[n == 0, 1, Sum[a[n-j]*Binomial[n-1, j-1]* If[j == 1, 0, (j-1)! - 2^(j-2)], {j, 1, n}]];
%t A187248 Table[a[n], {n, 0, 30}] (* _Jean-François Alcover_, Jun 16 2018, after _Alois P. Heinz_ *)
%Y A187248 Cf. A187247.
%K A187248 nonn
%O A187248 0,5
%A A187248 _Emeric Deutsch_, Mar 07 2011