OFFSET
0,2
COMMENTS
Bisection of A005981. The entries listed above suggest various congruences for a(n): a(n) = 1 (mod 10), a(n) = 1 + 70*n (mod 100), a(n) = 1 + 70*n + 200*n(n-1) (mod 1000). Are these congruences true for all n? For an arbitrary integer m, the sequence a(n) taken modulo m may eventually become periodic. Compare with A081727.
LINKS
Alois P. Heinz, Table of n, a(n) for n = 0..50
Christopher R. H. Hanusa, Alejandro H. Morales, and Martha Yip, Column convex matrices, G-cyclic orders, and flow polytopes, arXiv:2107.07326 [math.CO], 2021.
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+3))/(4n+3)! = (exp(2x)-2*sin(x)*exp(x)-1)/(2*exp(x)+cos(x)*(exp(2x)+1)). It appears that a(n) = (4n+3)!*coefficient of x^(4n+3) in the Taylor expansion of -4/(2*exp(x)+cos(x)*(exp(2x)+1)).
EXAMPLE
(1 4 5 3 2 6 7) is a 2 up, 2 down, 2 up permutation - one of the seventy-one permutations of this type in the symmetric group on 7 letters.
MAPLE
g:=(sinh(x)-sin(x))/(cos(x)*cosh(x)+1): series(%, x, 44):
seq(n!*coeff(%, x, n), n=3..45, 4); # Peter Luschny, Feb 07 2017
MATHEMATICA
Table[(CoefficientList[Series[(-Sin[x] + Sinh[x]) / (1 + Cos[x]*Cosh[x]), {x, 0, 60}], x] * Range[0, 59]!)[[n]], {n, 4, 60, 4}] (* Vaclav Kotesovec, Sep 09 2014 *)
CROSSREFS
KEYWORD
easy,nonn
AUTHOR
Peter Bala, Jul 13 2007
STATUS
approved