A131453 2 up, 2 down, ..., 2 up, 2 down permutations of length 4n+1.

1, 6, 1456, 2020656, 9336345856, 108480272749056, 2664103110372192256, 122840808510269863827456, 9758611490955498257378246656, 1251231616578606273788469919481856, 245996119743058288132230759497577005056, 71155698830255977656506481145458378597728256

Offset

0,2

Comments

Bisection of A005981.

Links

Alois P. Heinz, Table of n, a(n) for n = 0..50

L. Olivier, Bemerkungen über eine Art von Functionen, welche ähnliche Eigenschaften haben, wie der Cosinus und Sinus, J. Reine Angew. Math. 2 (1827), 243-251.

H. Prodinger and T. A. Tshifhumulo, On q-Olivier Functions, Annals of Combinatorics 6 (2002), 181-194.

B. Shapiro and A. Vainshtein, Counting real rational functions with all real critical values, arXiv:math/0209062 [math.AG], 2002.

B. Shapiro and A. Vainshtein, Counting real rational functions with all real critical values, Moscow Math. J. 3 (2003), 647-659.

Formula

E.g.f.: Sum_{n>=0} a(n)*(x^(4n+1))/(4n+1)! = (sin(x)*(exp(2x)+1)+cos(x)*(exp(2x)-1))/(2*exp(x)+cos(x)*(exp(2x)+1)).

Example

a(1) = 6. The six 2 up, 2 down permutations on 5 letters are (12543), (13542), (14532), (23541), (24532) and (34521).

Maple

g:= (tan(x)+exp(2*x)*(tan(x)+1)-1)/(exp(2*x)+2*exp(x)*sec(x)+1): series(%, x, 46):
seq(n!*coeff(%, x, n), n=1..45, 4); # Peter Luschny, Feb 07 2017

Mathematica

Table[(CoefficientList[Series[((-1 + E^(2*x))*Cos[x] + (1 + E^(2*x))*Sin[x]) / (2*E^x + (1 + E^(2*x))* Cos[x]), {x, 0, 80}], x] * Range[0, 77]!)[[n]], {n, 2, 78, 4}] (* Vaclav Kotesovec, Sep 09 2014 *)

Crossrefs

Cf. A000111, A005981, A058257, A131454, A131455.

Sequence in context: A300025 A107252 A358626 * A265862 A281255 A216934

Adjacent sequences: A131450 A131451 A131452 * A131454 A131455 A131456

Keyword

easy,nonn

Author

Peter Bala, Jul 13 2007

Status

approved