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).

1, 0, 0, 0, 2, 16, 104, 688, 5116, 44224, 438560, 4851136, 58603544, 764606016, 10715043104, 160692920256, 2570016145680, 43678554864128, 786135111482112, 14936420654180864, 298733557232591136, 6273502048592506112, 138018105454095739008, 3174423293668325353216

Offset: 0

Author

Emeric Deutsch, Mar 07 2011

Keywords

• nonn

Comments

a(n) = A187247(n,0).

Examples

			a(4)=2 because we have (1423) and (1324).
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..450

Crossrefs

Cf. A187247.

Programs

Maple

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);
# second Maple program:
a:= proc(n) option remember;
      `if`(n=0, 1, add(a(n-j)*binomial(n-1, j-1)*
      `if`(j=1, 0, (j-1)!-2^(j-2)), j=1..n))
    end:
seq(a(n), n=0..30);  # Alois P. Heinz, Apr 15 2017

Mathematica

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}]];
Table[a[n], {n, 0, 30}] (* Jean-François Alcover, Jun 16 2018, after Alois P. Heinz *)

Formula

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

a(n) ~ exp((-1-exp(2))/4) * n!. - Vaclav Kotesovec, Mar 18 2014