A187246 - OEIS

A187246

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

0, 0, 0, 1, 7, 42, 267, 1900, 15263, 137494, 1375195, 15127656, 181532895, 2359929682, 33039019643, 495585302836, 7929364861759, 134799202682670, 2426385648353595, 46101327318849376, 922026546377249663, 19362557473922767210, 425976264426301927195

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OFFSET

0,5

COMMENTS

a(n) = Sum_{k>=0} k*A187244(n,k).

LINKS

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

FORMULA

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

a(n) ~ (5/4-exp(1)+exp(2)/4) * n! = 0.378982196273617... * n!. - Vaclav Kotesovec, Mar 15 2014

D-finite with recurrence a(n) +(-n-3)*a(n-1) +(3*n-1)*a(n-2) +2*(-n+2)*a(n-3)=0. - R. J. Mathar, Jul 26 2022

EXAMPLE

a(4)=7 because each of the following permutations of {1,2,3,4} has 1 cycle with 2 alternating runs: (132)(4), (142)(3), (143)(2), (1)(243), (1243), (1342), and (1432); the remaining 17 permutations have none.

MAPLE

g := (1/4*(3+2*z+exp(2*z)-4*exp(z)))/(1-z): gser := series(g, z = 0, 25): seq(factorial(n)*coeff(gser, z, n), n = 0 .. 22);

# second Maple program:

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

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

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

end:

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

seq(a(n), n=0..30); # Alois P. Heinz, Apr 15 2017

MATHEMATICA

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

CROSSREFS