A187252 - OEIS

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).

%I #19 Jul 22 2022 12:00:32

%S 0,0,0,0,2,26,260,2508,25040,265552,3018144,36827872,481850240,

%T 6743052672,100629754112,1596624594688,26853667866624,477435143587840,

%U 8949520012611584,176443253945217024,3650510179312910336,79093615773747232768,1791150489194147512320

%N 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).

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

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

%F 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

%F 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

%e 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.

%p 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);

%o (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

%Y Cf. A187246, A187249, A187250.

%K nonn

%O 0,5

%A _Emeric Deutsch_, Mar 08 2011