A187252 Number of cycles with at least 3 alternating runs in all permutations of [n] (it is assumed that the smallest element of a cycle is in the first position).

0, 0, 0, 0, 2, 26, 260, 2508, 25040, 265552, 3018144, 36827872, 481850240, 6743052672, 100629754112, 1596624594688, 26853667866624, 477435143587840, 8949520012611584, 176443253945217024, 3650510179312910336, 79093615773747232768, 1791150489194147512320

Offset

0,5

Comments

a(n) = Sum_{k>=0} k * A187250(n,k).

Links

Table of n, a(n) for n=0..22.

Formula

E.g.f.: g(z) = -(1/4)*(2*z - 1 + exp(2*z) + 4*log(1-z)) / (1-z).

a(n) ~ n! * log(n) * (1 + (gamma - (1+exp(2))/4) / log(n)), where gamma is the Euler-Mascheroni constant A001620. - Vaclav Kotesovec, Dec 10 2021

Conjecture D-finite with recurrence a(n) +(-2*n-3)*a(n-1) +(n+7)*(n-1)*a(n-2) +4*(-n^2+2*n+1)*a(n-3) +4*(n-3)^2*a(n-4)=0. - R. J. Mathar, Jul 22 2022

Example

a(4) = 2 because among the permutations of {1,2,3,4} only 3421=(1324) and 4312=(1423) have cycles with more than 2 alternating runs.

Maple

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

Prog

(PARI) { my(z='z+O('z^33)); concat( [0, 0, 0, 0], Vec(serlaplace(-(1/4)*(2*z-1+exp(2*z)+4*log(1-z))/(1-z)))) } \\ Joerg Arndt, Apr 16 2017

Crossrefs

Cf. A187246, A187249, A187250.

Sequence in context: A198960 A289263 A296600 * A096233 A363985 A126673

Adjacent sequences: A187249 A187250 A187251 * A187253 A187254 A187255

Keyword

nonn

Author

Emeric Deutsch, Mar 08 2011

Status

approved