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

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

Cf. A187244.

Sequence in context: A152239 A152240 A221794 * A278152 A366222 A332082

Adjacent sequences: A187243 A187244 A187245 * A187247 A187248 A187249

Keyword

nonn

Author

Emeric Deutsch, Mar 07 2011

Status

approved