A246012 Number of permutations of n letters that have all valleys even and all peaks odd.

1, 1, 2, 4, 13, 50, 229, 1238, 7614, 52706, 405581, 3432022, 31684445, 316889938, 3413091138, 39387068998, 484828126705, 6340895228354, 87808618184425, 1283526229013174, 19749165195159006, 319067116612263218, 5400310536601145705, 95556575226489750694, 1764354398252386630937

Offset

0,3

Links

Vincenzo Librandi, Table of n, a(n) for n = 0..200

Ira M. Gessel and Yan Zhuang, Counting permutations by alternating descents, arXiv:1408.1886 [math.CO], 2014.

Y. Zhuang, Counting permutations by runs, J. Comb. Theory Ser. A 142 (2016), pp. 147-176.

Formula

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

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

Mathematica

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 *)

Crossrefs

Sequence in context: A290722 A341378 A058134 * A069730 A072605 A330344

Adjacent sequences: A246009 A246010 A246011 * A246013 A246014 A246015

Keyword

nonn,easy

Author

N. J. A. Sloane, Aug 11 2014

Status

approved