A187245 Number of permutations of [n] having no cycle with 2 alternating runs (it is assumed that the smallest element of the cycle is in the first position).

1, 1, 2, 5, 17, 78, 463, 3315, 27164, 247975, 2492539, 27422698, 328607417, 4266367567, 59686293284, 895068242601, 14320843215019, 243467476610732, 4382635181281015, 83272415871044649, 1665465961530365026, 34974843092354081119, 769445564105823722109

Offset

0,3

Comments

a(n) = A187244(n,0).

Links

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

Formula

E.g.f.: g(z)=exp[(4exp(z)-exp(2z)-3-2z)/4]/(1-z).

a(n) ~ exp(exp(1)-exp(2)/4-5/4) * n! = 0.68455780023755436... * n!. - Vaclav Kotesovec, Mar 15 2014

Example

a(3)=5 because we have among the 6 permutations of {1,2,3} only 312=(132) has a cycle with 2 alternating runs.

Maple

g := exp((4*exp(z)-exp(2*z)-3-2*z)*1/4)/(1-z): gser := series(g, z = 0, 25): seq(factorial(n)*coeff(gser, z, n), n = 0 .. 22);
# second Maple program:
a:= proc(n) option remember;
      `if`(n=0, 1, add(a(n-j)*binomial(n-1, j-1)*
      `if`(j=1, 1, (j-1)!-(2^(j-2)-1)), j=1..n))
    end:
seq(a(n), n=0..30);  # Alois P. Heinz, Apr 15 2017

Mathematica

CoefficientList[Series[E^((4*E^x-E^(2*x)-3-2*x)/4)/(1-x), {x, 0, 20}], x]* Range[0, 20]! (* Vaclav Kotesovec, Mar 15 2014 *)

Crossrefs

Cf. A187244.

Sequence in context: A184509 A020096 A362109 * A361243 A302194 A289739

Adjacent sequences: A187242 A187243 A187244 * A187246 A187247 A187248

Keyword

nonn

Author

Emeric Deutsch, Mar 07 2011

Status

approved