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

Author

Emeric Deutsch, Mar 07 2011

Keywords

• nonn

Comments

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

Examples

			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.
		

Links

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

Crossrefs

Cf. A187244.

Programs

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 *)

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