A186369 Number of excedances in all cycle-up-down permutations of {1,2,...,n}. A permutation is said to be cycle-up-down if it is a product of up-down cycles. A cycle (b(1), b(2), ...) is said to be up-down if, when written with its smallest element in the first position, it satisfies b(1) < b(2) > b(3) < ... .

0, 0, 1, 4, 20, 100, 568, 3480, 23552, 172280, 1369600, 11687996, 107154944, 1046876220, 10891574272, 119977808752, 1397745975296, 17147547551920, 221248695107584, 2992325932512948, 42370119842398208, 626422742409351380, 9659199474524225536, 154997830582262971784

Offset

0,4

Comments

a(n) = Sum_{k>=0} k*A186368(n,k).

Links

Table of n, a(n) for n=0..23.

E. Deutsch and S. Elizalde, Cycle up-down permutations, arXiv:0909.5199 [math.CO], 2009.

Formula

E.g.f.: [z(1+sin z)-cos z * log(sec z + tan z)]/[2 cos z *(1-sin z)].

a(n) ~ n! * n^2 * (2/Pi)^(n+2) * (1 - log(n)/n). - Vaclav Kotesovec, Aug 23 2014

Example

a(3) = 4 because the permutations (1)(2)(3), (1)(23), (12)(3), (13)(2), (132) have a total of 0+1+1+1+1=4 excedances.

Maple

g := ((z*(1+sin(z))-cos(z)*ln(sec(z)+tan(z)))*1/2)/(cos(z)*(1-sin(z))): gser := series(g, z = 0, 30): seq(factorial(n)*coeff(gser, z, n), n = 0 .. 22);

Mathematica

T[n_, k_] := n! SeriesCoefficient[(Sec[z Sqrt[t]]+ Tan[z Sqrt[t]])^( 1/Sqrt[t])/Cos[z Sqrt[t]], {z, 0, n}, {t, 0, k}];
Table[Sum[k T[n, k], {k, 0, n/2}], {n, 0, 22}] (* Jean-François Alcover, Aug 07 2018 *)

Crossrefs

Cf. A186368.

Sequence in context: A216099 A105480 A242156 * A093440 A168606 A229135

Adjacent sequences: A186366 A186367 A186368 * A186370 A186371 A186372

Keyword

nonn

Author

Emeric Deutsch, Feb 28 2011

Status

approved