A246012 - OEIS

%I #19 Sep 05 2021 05:41:50

%S 1,1,2,4,13,50,229,1238,7614,52706,405581,3432022,31684445,316889938,

%T 3413091138,39387068998,484828126705,6340895228354,87808618184425,

%U 1283526229013174,19749165195159006,319067116612263218,5400310536601145705,95556575226489750694,1764354398252386630937

%N Number of permutations of n letters that have all valleys even and all peaks odd.

%H Vincenzo Librandi, <a href="/A246012/b246012.txt">Table of n, a(n) for n = 0..200</a>

%H Ira M. Gessel and Yan Zhuang, <a href="http://arxiv.org/abs/1408.1886">Counting permutations by alternating descents</a>, arXiv:1408.1886 [math.CO], 2014.

%H Y. Zhuang, <a href="https://doi.org/10.1016/j.jcta.2016.04.002">Counting permutations by runs</a>, J. Comb. Theory Ser. A 142 (2016), pp. 147-176.

%F E.g.f.: (3*sin(x/2)+3*cosh(sqrt(3)*x/2))/(3*cos(x/2)-sqrt(3)*sinh(sqrt(3)*x/2)).

%F a(n) ~ n! / (2^n * r^(n+1)), where r = 0.649914158110577478699523... is the root of the equation 6*cos(r) - sqrt(3)*exp(sqrt(3)*r) + sqrt(3)*exp(-sqrt(3)*r) = 0. - _Vaclav Kotesovec_, Aug 29 2014

%t CoefficientList[Series[(3*Cosh[(Sqrt[3]*x)/2] + 3*Sin[x/2]) / (3*Cos[x/2] - Sqrt[3]*Sinh[(Sqrt[3]*x)/2]), {x, 0, 20}], x] * Range[0, 20]! (* _Vaclav Kotesovec_, Aug 29 2014 *)

%K nonn,easy

%O 0,3

%A _N. J. A. Sloane_, Aug 11 2014