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

Author

Emeric Deutsch, Mar 08 2011

Keywords

• nonn

Comments

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

Examples

			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.
		

Crossrefs

Cf. A187246, A187249, A187250.

Programs

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);

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

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