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
KEYWORD
easy,nonn
AUTHOR
Peter Bala, Jul 13 2007
STATUS
approved